13 files changed, 14431 insertions, 0 deletions

diff --git a/šola/ana2/fja.scad b/šola/ana2/fja.scad
new file mode 100644
index 0000000..4d18a17
--- /dev/null
+++ b/šola/ana2/fja.scad
@@ -0,0 +1,13 @@
+echo(version=version());
+function sinh(x) = (exp(1)-exp(-x))/2;
+function f(x, y, z) = sinh(z)*sinh(y)-sin(x);
+epsilon = 0.01;
+rob = 1;
+korak = 0.1;
+particle = 0.1;
+for (x = [-rob : korak : rob])
+    for (y = [-rob : korak : rob])
+        for (z = [-rob : korak : rob])
+            if (f(x, y, z) < epsilon)
+                translate([x, y, z])
+                    cube(particle);
\ No newline at end of file
diff --git a/šola/ana2/kolokvij1.lyx b/šola/ana2/kolokvij1.lyx
new file mode 100644
index 0000000..a4f9569
--- /dev/null
+++ b/šola/ana2/kolokvij1.lyx
@@ -0,0 +1,976 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\usepackage{amsmath}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\DeclareMathOperator{\g}{g}
+\DeclareMathOperator{\sled}{sled}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Cir}{Cir}
+\DeclareMathOperator{\ecc}{ecc}
+\DeclareMathOperator{\rad}{rad}
+\DeclareMathOperator{\diam}{diam}
+\newcommand\euler{e}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+\end_modules
+\maintain_unincluded_children false
+\language slovene
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics xetex
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1cm
+\topmargin 1cm
+\rightmargin 1cm
+\bottommargin 2cm
+\headheight 1cm
+\headsep 1cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+setlength{
+\backslash
+columnseprule}{0.2pt}
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Podmnožice v evklidskih prostorih
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ zaprta, če 
+\begin_inset Formula $\forall$
+\end_inset
+
+ zaporedje s členi v 
+\begin_inset Formula $A:$
+\end_inset
+
+ vsa stekališča v 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ kompaktna, če 
+\begin_inset Formula $\forall$
+\end_inset
+
+ zaporedje s členi v 
+\begin_inset Formula $A$
+\end_inset
+
+:
+\begin_inset Formula $\exists$
+\end_inset
+
+ stekališče v 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f$
+\end_inset
+
+ je kompozitum zveznih 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ zvezna
+\end_layout
+
+\begin_layout Standard
+za 
+\begin_inset Formula $x\in\mathbb{R}^{k}$
+\end_inset
+
+: 
+\begin_inset Formula $\text{\left|\left|x\right|\right|\ensuremath{\coloneqq}}\sqrt{x_{1}^{2}+\cdots+x_{k}^{2}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A\subseteq\mathbb{R}^{k}$
+\end_inset
+
+ omejena 
+\begin_inset Formula $\Leftrightarrow\exists M\in\mathbb{R}\forall x\in A:\left|\left|x\right|\right|<M$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K\left(s\in\mathbb{R}^{k},M\in\mathbb{R}\right)\coloneqq\left\{ s+x\in\mathbb{R}^{k};\text{\left|\left|x\right|\right|}<M\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $x\in A$
+\end_inset
+
+ notranja 
+\begin_inset Formula $\Leftrightarrow\exists\varepsilon\ni:K\left(x,\varepsilon\right)\subset A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A\subseteq\mathbb{R}^{k}$
+\end_inset
+
+ odprta 
+\begin_inset Formula $\Leftrightarrow\forall x\in A:x$
+\end_inset
+
+ notranja
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A\subseteq\mathbb{R}^{k}$
+\end_inset
+
+ zaprta 
+\begin_inset Formula $\Leftrightarrow\mathbb{R}^{k}\setminus A$
+\end_inset
+
+ odprta
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $x\in\mathbb{R}^{k}$
+\end_inset
+
+ stekališče 
+\begin_inset Formula $A\Leftrightarrow\forall\varepsilon>0:K\left(x,\varepsilon\right)\cup\left(A\setminus\left\{ x\right\} \right)\not=\emptyset$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $x\in\mathbb{R}^{k}$
+\end_inset
+
+ izolirana točka 
+\begin_inset Formula $A\Leftrightarrow x$
+\end_inset
+
+ ni stekališče 
+\begin_inset Formula $A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Zaporedje 
+\begin_inset Formula $a_{n}:\mathbb{N}\to\mathbb{R}^{k}$
+\end_inset
+
+ konvergira proti 
+\begin_inset Formula $a\in\mathbb{R}^{k}$
+\end_inset
+
+ kadar 
+\begin_inset Formula $\forall\varepsilon>0\exists N\in\mathbb{N}\forall n>N:\left|\left|a_{n}-a\right|\right|<\varepsilon$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $s\in\mathbb{R}^{k}$
+\end_inset
+
+ stekališče zap.
+ 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ v 
+\begin_inset Formula $\text{\ensuremath{\varepsilon}}-$
+\end_inset
+
+okolici 
+\begin_inset Formula $s$
+\end_inset
+
+ je 
+\begin_inset Formula $\infty$
+\end_inset
+
+mnogo členov
+\end_layout
+
+\begin_layout Standard
+Vsako omejeno zaporedje ima stekališče.
+\end_layout
+
+\begin_layout Section
+Funkcije več spremenljivk
+\end_layout
+
+\begin_layout Standard
+fja 
+\begin_inset Formula $k$
+\end_inset
+
+ spremenljivk je preslikava 
+\begin_inset Formula $f:D\subseteq\mathbb{R}^{k}\to\mathbb{R}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $L\in\mathbb{R}$
+\end_inset
+
+ je limita 
+\begin_inset Formula $f:D\subseteq\text{\ensuremath{\mathbb{R}^{k}}}\to\mathbb{R}$
+\end_inset
+
+ v stekališču 
+\begin_inset Formula $a\in D$
+\end_inset
+
+, če 
+\begin_inset Formula $\forall\varepsilon>0\exists\delta>0\forall x\in D\setminus\left\{ a\right\} :\left|\left|x-a\right|\right|<\delta\Rightarrow\left|f\left(x\right)-L\right|<\varepsilon$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\lim_{x\to a}\left(f\oslash g\right)x=\lim_{x\to a}fx\oslash\lim_{x\to a}gx$
+\end_inset
+
+, če obstajata.
+ 
+\begin_inset Formula $\oslash\in\left\{ +,-,\cdot\right\} $
+\end_inset
+
+.
+ 
+\begin_inset Formula $\oslash$
+\end_inset
+
+ je lahko deljenje, kadar 
+\begin_inset Formula $\lim_{x\to a}gx\not=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f$
+\end_inset
+
+ zvezna v 
+\begin_inset Formula $a\Leftrightarrow\forall\varepsilon\exists\delta\forall x\in D:\left|\left|x-a\right|\right|<\delta\Rightarrow\left|fx-fa\right|<\varepsilon$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f$
+\end_inset
+
+ zvezna 
+\begin_inset Formula $\Leftrightarrow\forall a\in D:f$
+\end_inset
+
+ zvezna v 
+\begin_inset Formula $a$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f$
+\end_inset
+
+ zvezna v stekališču 
+\begin_inset Formula $a\Leftrightarrow\lim_{x\to a}fx=fa$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A\subseteq\mathbb{R}^{k}$
+\end_inset
+
+ kompaktna, 
+\begin_inset Formula $f:A\to\mathbb{R}$
+\end_inset
+
+ zvezna
+\begin_inset Formula $\Rightarrow f$
+\end_inset
+
+ omejena in doseže maksimum in minimum (obstoj globalnega ekstrema).
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f,g$
+\end_inset
+
+ zv.
+ 
+\begin_inset Formula $\Rightarrow f\oslash g$
+\end_inset
+
+ zv.
+ 
+\begin_inset Formula $\oslash\in\left\{ +,-,\cdot,\circ\right\} ,\oslash=/\Leftrightarrow\forall x:gx\not=0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Odvodi funkcij več spremenljivk
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f_{x_{i}}a$
+\end_inset
+
+, 
+\begin_inset Formula $i\in\left\{ 1..k\right\} $
+\end_inset
+
+ je odvod fje 
+\begin_inset Formula $x_{i}\to f\left(a_{1},\dots,a_{i-1},x_{i},a_{i+1},\dots,a_{k}\right)$
+\end_inset
+
+ v točki 
+\begin_inset Formula $a_{i}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $f_{x_{i}}a=\lim_{x_{i}\to a_{i}}\frac{f\left(a_{1},\dots,a_{i-1},x_{i},a_{i+1},\dots,a_{k}\right)-fa}{x_{i}-a_{i}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Tang.
+ ravn.
+ v 
+\begin_inset Formula $a=\left(b,c\right)$
+\end_inset
+
+
+\begin_inset Formula $a=\left(b,c\right)$
+\end_inset
+
+: 
+\begin_inset Formula $z=fa+f_{x}a\cdot\left(x-b\right)+f_{y}a\cdot\left(y-c\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f$
+\end_inset
+
+ je odvedljiva v 
+\begin_inset Formula $a\Leftrightarrow\lim_{h\to\left(0,0\right)}\frac{R_{a}\left(a+h\right)}{\left|\left|h\right|\right|}=0$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $R_{a}\left(a+h\right)\coloneqq f\left(a+h\right)-fa-f_{x}\left(a\right)\cdot u+f_{y}\left(a\right)\cdot v$
+\end_inset
+
+ za 
+\begin_inset Formula $h=\left(u,v\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $dfa\coloneqq\left[f_{x_{1}}a\cdots f_{x_{k}}a\right]=\nabla fa,dfa\cdot h\coloneqq f_{x_{1}}a\cdot h_{1}+\cdots+f_{x_{k}}a\cdot h_{k}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f$
+\end_inset
+
+ v 
+\begin_inset Formula $a$
+\end_inset
+
+ odvedljiva 
+\begin_inset Formula $\Rightarrow f$
+\end_inset
+
+ v 
+\begin_inset Formula $a$
+\end_inset
+
+ zvezna
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall i\in\left\{ 1..k\right\} :\exists f_{x_{i}}\wedge f_{x_{i}}$
+\end_inset
+
+ zvezna 
+\begin_inset Formula $\Rightarrow f$
+\end_inset
+
+ odvedljiva
+\end_layout
+
+\begin_layout Standard
+Lagrange: 
+\begin_inset Formula $fx_{1}-fx_{2}=f'\xi\left(x_{2}-x_{1}\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f\in C^{r}U\sim$
+\end_inset
+
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ 
+\begin_inset Formula $r-$
+\end_inset
+
+krat zvezno odvedljiva v vsaki točki 
+\begin_inset Formula $U$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f:U\subseteq\mathbb{R}^{k}\to\mathbb{R},f\in C^{2}U\Rightarrow\forall i,j\in\left\{ 1..k\right\} :f_{x_{i}}=f_{x_{j}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\frac{df}{d\vec{s}}\left(x,y\right)=\lim_{t\to0}\frac{f((x,y)+t\vec{s})-f(x,y)}{t}=s_{1}f(x,y)+s_{2}f_{x}(x,y)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Tangentna ravnina v 
+\begin_inset Formula $\left(x_{0},y_{0},f\left(x_{0},y_{0}\right)\right)$
+\end_inset
+
+ je 
+\begin_inset Formula $f_{x}\left(x_{0},y_{0}\right)\left(x-x_{0}\right)+f_{y}\left(x_{0},y_{0}\right)\left(y-y_{0}\right)-z+f\left(x_{0},y_{0}\right)$
+\end_inset
+
+ in razpenjata jo vektorja 
+\begin_inset Formula $\left(1,0,f_{x}\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $\left(0,1,f_{y}\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Taylorjeva formula in verižno pravilo
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $f$
+\end_inset
+
+ diferenciabilna v 
+\begin_inset Formula $a\in\mathbb{R}^{k}\Rightarrow f\left(a+h\right)\cong fa+dfa\cdot h=fa+f_{x_{1}}a\cdot h_{1}+\cdots+f_{x_{k}}a\cdot h_{k}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $U\subseteq R^{k}$
+\end_inset
+
+ odprta in 
+\begin_inset Formula $f:U\to\mathbb{R},f\in C^{n+1}U$
+\end_inset
+
+.
+ Naj bo 
+\begin_inset Formula $D_{f,r,a}$
+\end_inset
+
+ vektor vseh parcialnih odvodov reda 
+\begin_inset Formula $r$
+\end_inset
+
+ v točki 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Primer: 
+\begin_inset Formula $D_{f,2,a}=\left(f_{xx}a,2f_{xy}a+f_{yy}a\right)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $D_{f,0,a}\coloneqq f\left(a\right)$
+\end_inset
+
+.
+ Naj bo 
+\begin_inset Formula $H_{r}$
+\end_inset
+
+ vektor z vsemi kombinacijami dolžine 
+\begin_inset Formula $r$
+\end_inset
+
+ komponent 
+\begin_inset Formula $h$
+\end_inset
+
+.
+ Primer: 
+\begin_inset Formula $H_{2}=\left(uu,2uv,vv\right)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $H_{0}=1$
+\end_inset
+
+.
+ 
+\begin_inset Formula $D_{f,r,a}\cdot H_{r}$
+\end_inset
+
+ je njun skalarni produkt.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+T_{f,a,n}\left(h_{1}=x-a,h_{2}=y-b\right)=\sum_{i=0}^{n}\frac{1}{i!}\left(D_{f,i,a}\cdot H_{i}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Ekstremalni problemi
+\end_layout
+
+\begin_layout Standard
+Kandidati so 
+\begin_inset Formula $a$
+\end_inset
+
+, da 
+\begin_inset Formula $\nabla fa=0$
+\end_inset
+
+ ali 
+\begin_inset Formula $f$
+\end_inset
+
+ ni odv.
+ v 
+\begin_inset Formula $a$
+\end_inset
+
+ ali 
+\begin_inset Formula $a$
+\end_inset
+
+ robna točka.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+H\left(a,b\right)=\left[\begin{array}{cc}
+f_{xx}\left(a,b\right) & f_{xy}\left(a,b\right)\\
+f_{yx}\left(a,b\right) & f_{yy}\left(a,b\right)
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det H\left(a,b\right)>0$
+\end_inset
+
+: 
+\begin_inset Formula $f_{xx}\left(a,b\right)>0$
+\end_inset
+
+ l.
+ min., 
+\begin_inset Formula $f_{xx}\left(a,b\right)<0$
+\end_inset
+
+ l.
+ max.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det H\left(a,b\right)<0$
+\end_inset
+
+ sedlo
+\end_layout
+
+\begin_layout Standard
+Izrek o implicitni funkciji: 
+\begin_inset Formula $D\subseteq\mathbb{R}^{2}$
+\end_inset
+
+ odprta, 
+\begin_inset Formula $f:D\to\mathbb{R}$
+\end_inset
+
+ zvezno parcialno odvedljiva.
+ 
+\begin_inset Formula $K=\left\{ \left(x,y\right)\in D;f\left(x,y\right)=0\right\} $
+\end_inset
+
+.
+ Za 
+\begin_inset Formula $\left(a,b\right)\in D$
+\end_inset
+
+, 
+\begin_inset Formula $f\left(a,b\right)=0\wedge\nabla f\left(a,b\right)\not=0\exists h\left(a\right)=b,f\left(x,h\left(x\right)\right)=0\forall x\in U$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Vezani ekstrem: 
+\begin_inset Formula $D^{\text{odp.}}\subseteq\mathbb{R}^{n}$
+\end_inset
+
+, 
+\begin_inset Formula $f:D\to\mathbb{R}$
+\end_inset
+
+ diferenciabilna na 
+\begin_inset Formula $D$
+\end_inset
+
+.
+ let 
+\begin_inset Formula $g:D\to\mathbb{R}$
+\end_inset
+
+ zvezno parcialno odvedljiva, 
+\begin_inset Formula $A\coloneqq\left\{ x\in D;gx=0\right\} $
+\end_inset
+
+.
+ 
+\begin_inset Formula $\exists$
+\end_inset
+
+ vezani ekstrem 
+\begin_inset Formula $f$
+\end_inset
+
+ pri pogoju 
+\begin_inset Formula $g\Leftrightarrow\nabla fa=\lambda\nabla ga$
+\end_inset
+
+.
+ Kandidati za vezane ekstreme so stac.
+ točke fje 
+\begin_inset Formula $F\left(x,\lambda\right)=fx-\lambda gx$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Krivulje in ploskve
+\end_layout
+
+\begin_layout Standard
+Pot v 
+\begin_inset Formula $\mathbb{R}^{3}\sim\vec{r}:I\to\mathbb{R}^{3},I\in\mathbb{R}$
+\end_inset
+
+ interval.
+ 
+\begin_inset Formula $\forall t\in I:\vec{r}t=\left(xt,yt,zt\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Odvod poti: 
+\begin_inset Formula $\dot{\vec{r}}\left(t\right)=\left(\dot{x}\left(t\right),\dot{y}\left(t\right),\dot{z}\left(t\right)\right)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\dot{\vec{r}}t$
+\end_inset
+
+ je tangentni vektor na krivuljo v točki 
+\begin_inset Formula $\vec{r}t$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dolžina poti 
+\begin_inset Formula $\vec{r}:\left[a,b\right]\to\mathbb{R}^{3}$
+\end_inset
+
+ je 
+\begin_inset Formula 
+\[
+L=\int_{a}^{b}\left|\dot{\vec{r}}t\right|dt=\int_{a}^{b}\sqrt{\dot{x}^{2}t+\dot{y}^{2}t}dt
+\]
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Ploščina območja, ki ga omejuje krivulja, če je parametrizacija taka, da
+ je krivulja levo od 
+\begin_inset Formula $\vec{r}t$
+\end_inset
+
+: 
+\begin_inset Formula $\text{Pl\left(D\right)=\ensuremath{\frac{1}{2}\int_{a}^{b}\left(xt\dot{y}t-\dot{x}tyt\right)dt}}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Ploskev eksplicitno kot graf 
+\begin_inset Formula $f:D^{\text{odp.}}\subseteq\mathbb{R}^{2}\to\mathbb{R}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ difer.
+ v 
+\begin_inset Formula $\left(x,y\right)\Rightarrow$
+\end_inset
+
+ v 
+\begin_inset Formula $\left(x,y,f\left(x,y\right)\right)$
+\end_inset
+
+ definiramo tangentno ravnino z normalo 
+\begin_inset Formula $\left(-f_{x}\left(x,y\right),-f_{y}\left(x,y\right),1\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ploskev implicitno: 
+\begin_inset Formula $f:\mathbb{R}^{3}\to\mathbb{R}$
+\end_inset
+
+ zvezno parcialno odvedljiva.
+\begin_inset Formula 
+\[
+P=\left\{ \left(x,y,z\right)\in\mathbb{R}^{3};f\left(x,y,z\right)=0\right\} 
+\]
+
+\end_inset
+
+Če 
+\begin_inset Formula $\forall\left(x,y,z\right)\in P:\nabla f\left(x,y,z\right)\not=0\Rightarrow P$
+\end_inset
+
+ ploskev v 
+\begin_inset Formula $\mathbb{R}^{3}$
+\end_inset
+
+, saj je po izreku o implicitni fji 
+\begin_inset Formula $P$
+\end_inset
+
+ lokalno graf fje dveh spremenljivk.
+ Normala tangentne ravnine v 
+\begin_inset Formula $\left(x,y,z\right)\in P$
+\end_inset
+
+ ima normalo 
+\begin_inset Formula $\nabla f\left(x,y,z\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/ars/ol2.odt b/šola/ars/ol2.odt
new file mode 100644
index 0000000..32a2d7d
--- /dev/null
+++ b/šola/ars/ol2.odt
diff --git a/šola/ds2/kolokvij2.lyx b/šola/ds2/kolokvij2.lyx
new file mode 100644
index 0000000..0d2e149
--- /dev/null
+++ b/šola/ds2/kolokvij2.lyx
@@ -0,0 +1,3368 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\DeclareMathOperator{\g}{g}
+\DeclareMathOperator{\sled}{sled}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\Cir}{Cir}
+\DeclareMathOperator{\ecc}{ecc}
+\DeclareMathOperator{\rad}{rad}
+\DeclareMathOperator{\diam}{diam}
+\newcommand\euler{e}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+\end_modules
+\maintain_unincluded_children false
+\language slovene
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics xetex
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1cm
+\topmargin 1cm
+\rightmargin 1cm
+\bottommargin 2cm
+\headheight 1cm
+\headsep 1cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+setlength{
+\backslash
+columnseprule}{0.2pt}
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Največja stopnja 
+\begin_inset Formula $\Delta G$
+\end_inset
+
+, najmanjša 
+\begin_inset Formula $\delta G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Rokovanje: 
+\begin_inset Formula $\sum_{v\in VG}\deg_{G}v=2\left|EG\right|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Vsak graf ima sodo mnogo vozlišč lihe stopnje.
+\end_layout
+
+\begin_layout Standard
+Presek/unija grafov je presek/unija njunih 
+\begin_inset Formula $V$
+\end_inset
+
+ in 
+\begin_inset Formula $E$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G\cup H$
+\end_inset
+
+ je disjunktna unija grafov 
+\begin_inset Formula $\sim$
+\end_inset
+
+ 
+\begin_inset Formula $VG\cap VH=\emptyset$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Komplement grafa: 
+\begin_inset Formula $\overline{G}$
+\end_inset
+
+ (obratna povezanost)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $0\leq\left|EG\right|\leq{\left|VG\right| \choose 2}\quad\quad\quad$
+\end_inset
+
+Za padajoče 
+\begin_inset Formula $d_{i}$
+\end_inset
+
+ velja:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(d_{1},\dots d_{n}\right)$
+\end_inset
+
+ graf 
+\begin_inset Formula $\Leftrightarrow\left(d_{2}-1,\dots,d_{d_{1}+1}-1,\dots,d_{n}\right)$
+\end_inset
+
+ graf
+\end_layout
+
+\begin_layout Standard
+Če je 
+\begin_inset Formula $AG$
+\end_inset
+
+ matrika sosednosti, 
+\begin_inset Formula $\left(\left(AG\right)^{n}\right)_{i,j}$
+\end_inset
+
+ pove št.
+ 
+\begin_inset Formula $i,j-$
+\end_inset
+
+poti.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\text{Število trikotnikov: }\frac{\sled\left(\left(AG\right)^{3}\right)}{2\cdot3}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left|EK_{n}\right|={n \choose 2}$
+\end_inset
+
+, 
+\begin_inset Formula ${n \choose k}=\frac{n!}{k!\left(n-k\right)!}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Sprehod
+\end_layout
+
+\begin_layout Standard
+je zaporedje vozlišč, ki so verižno povezana.
+\end_layout
+
+\begin_layout Standard
+Dolžina sprehoda je število prehojenih povezav.
+\end_layout
+
+\begin_layout Standard
+Sklenjen sprehod dolžine 
+\begin_inset Formula $k$
+\end_inset
+
+: 
+\begin_inset Formula $v_{0}=v_{k}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Enostaven sprehod ima disjunktna vozlišča razen 
+\begin_inset Formula $\left(v_{0},v_{k}\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Pot v grafu: podgraf 
+\begin_inset Formula $P_{k}$
+\end_inset
+
+ 
+\begin_inset Formula $\sim$
+\end_inset
+
+ enostaven nesklenjen sprehod.
+\end_layout
+
+\begin_layout Paragraph
+Cikel
+\end_layout
+
+\begin_layout Standard
+podgraf, ki je enostaven sklenjen sprehod dolžine 
+\begin_inset Formula $>3$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Če v 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $\exists$
+\end_inset
+
+ dve različni 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ premore cikel
+\end_layout
+
+\begin_layout Standard
+Sklenjen sprehod lihe dolžine 
+\begin_inset Formula $\in G$
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ lih cikel 
+\begin_inset Formula $\in G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Povezanost
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $u,v$
+\end_inset
+
+ sta v isti komponenti 
+\begin_inset Formula $\text{\ensuremath{\sim}}$
+\end_inset
+
+ 
+\begin_inset Formula $\text{\ensuremath{\exists}}$
+\end_inset
+
+ 
+\begin_inset Formula $u,v-$
+\end_inset
+
+pot
+\end_layout
+
+\begin_layout Standard
+Število komponent grafa: 
+\begin_inset Formula $\Omega G$
+\end_inset
+
+.
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\sim\Omega G=1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Komponenta je maksimalen povezan podgraf.
+\end_layout
+
+\begin_layout Standard
+Premer: 
+\begin_inset Formula $\diam G=\max\left\{ d_{G}\left(v,u\right);\forall v,u\in VG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ekscentričnost: 
+\begin_inset Formula $\ecc_{G}u=max\left\{ d_{G}\left(u,x\right);\forall x\in VG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Polmer: 
+\begin_inset Formula $\rad G=\min\left\{ \ecc u;\forall u\in VG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\diam C_{n}=\rad C_{n}=\lfloor\frac{n}{2}\rfloor$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+(Liha) ožina (
+\begin_inset Formula $\g G$
+\end_inset
+
+) je dolžina najkrajšega (lihega) cikla.
+\end_layout
+
+\begin_layout Standard
+Vsaj en od 
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $\overline{G}$
+\end_inset
+
+ je povezan.
+\end_layout
+
+\begin_layout Standard
+Povezava 
+\begin_inset Formula $e$
+\end_inset
+
+ je most 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $\Omega\left(G-e\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $u$
+\end_inset
+
+ je prerezno vozlišče 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $\Omega\left(G-u\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za nepovezan 
+\begin_inset Formula $G$
+\end_inset
+
+ velja 
+\begin_inset Formula $\diam\overline{G}\leq2$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Dvodelni
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sim V=A\cup B,A\cap B=\emptyset,\forall uv\in E:u\in A\oplus v\in A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{m,n}$
+\end_inset
+
+ je poln dvodelni graf, 
+\begin_inset Formula $\left|A\right|=m$
+\end_inset
+
+, 
+\begin_inset Formula $\left|B\right|=n$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen 
+\begin_inset Formula $\Leftrightarrow\forall$
+\end_inset
+
+ komponenta 
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelna
+\end_layout
+
+\begin_layout Standard
+Pot, sod cikel, hiperkocka so dvodelni, petersenov ni.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen 
+\begin_inset Formula $\Leftrightarrow G$
+\end_inset
+
+ ne vsebuje lihega cikla.
+\end_layout
+
+\begin_layout Standard
+Biparticija glede na parnost 
+\begin_inset Formula $d_{G}\left(u,x_{0}\right)$
+\end_inset
+
+, 
+\begin_inset Formula $x_{0}$
+\end_inset
+
+ fiksen.
+\end_layout
+
+\begin_layout Standard
+Dvodelen 
+\begin_inset Formula $k-$
+\end_inset
+
+regularen, 
+\begin_inset Formula $\left|E\right|\ge1\Rightarrow$
+\end_inset
+
+
+\begin_inset Formula $\left|A\right|=\left|B\right|$
+\end_inset
+
+.
+ Dokaz: 
+\begin_inset Formula $\sum_{u\in A}\deg u=\left|E\right|=\cancel{k}\left|A\right|=\cancel{k}\left|B\right|=\sum_{u\in B}\deg u$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Krožni
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Cir\left(n,\left\{ s_{1},\dots,s_{k}\right\} \right):\left|V\right|=n,$
+\end_inset
+
+ množica preskokov
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Omega\Cir\left(n,\left\{ s,n-s\right\} \right)=\gcd\left\{ n,s\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $W_{n}$
+\end_inset
+
+ (kolo) pa je cikel z univerzalnim vozliščem.
+\end_layout
+
+\begin_layout Paragraph
+Homomorfizem
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\varphi:VG\to VH$
+\end_inset
+
+ slika povezave v povezave
+\end_layout
+
+\begin_layout Standard
+Primer: 
+\begin_inset Formula $K_{2}$
+\end_inset
+
+ je homomorfna slika 
+\begin_inset Formula $\forall$
+\end_inset
+
+ bipartitnega grafa.
+\end_layout
+
+\begin_layout Standard
+V povezavah in vozliščih surjektiven 
+\begin_inset Formula $hm\varphi$
+\end_inset
+
+ je epimorfizem.
+\end_layout
+
+\begin_layout Standard
+V vozliščih injektiven 
+\begin_inset Formula $hm\varphi$
+\end_inset
+
+ je monomorfizem/vložitev.
+\end_layout
+
+\begin_layout Standard
+Vložitev, ki ohranja razdalje, je izometrična.
+\end_layout
+
+\begin_layout Standard
+Kompozitum homomorfizmov je spet homomorfizem.
+\end_layout
+
+\begin_layout Standard
+Izomorfizem je bijektivni 
+\begin_inset Formula $hm\varphi$
+\end_inset
+
+ s homomorfnim inverzom.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $im\varphi$
+\end_inset
+
+ 
+\begin_inset Formula $f:VG\to VH$
+\end_inset
+
+ 
+\begin_inset Formula $\forall u,v\in VG:uv\in EG\Leftrightarrow fufv\in EH$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Nad množico vseh grafov 
+\begin_inset Formula $\mathcal{G}$
+\end_inset
+
+ je izomorfizem (
+\begin_inset Formula $\cong$
+\end_inset
+
+) ekv.
+ rel.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $im\varphi$
+\end_inset
+
+ 
+\begin_inset Formula $G\to G$
+\end_inset
+
+ je avtomorfizem.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Aut G$
+\end_inset
+
+ je grupa avtomorfizmov s komponiranjem.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Aut K_{n}=\left\{ \pi\in S_{n}=\text{permutacije }n\text{ elementov}\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Aut P_{n}=\left\{ \text{trivialni }id,f\left(i\right)=n-i-1\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $\Aut G\cong\Aut\overline{G}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izomorfizem ohranja stopnje, št.
+ 
+\begin_inset Formula $C_{4}$
+\end_inset
+
+, povezanost, 
+\begin_inset Formula $\left|EG\right|,\dots$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G\cong\overline{G}\Rightarrow\left|VG\right|\%4\in\left\{ 0,1\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Operacije
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ vpeti podgraf 
+\begin_inset Formula $G\Leftrightarrow\exists F\subseteq EG\ni:H=G-F$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ inducirani podgraf 
+\begin_inset Formula $G\Leftrightarrow\exists S\subseteq VG\ni:H=G-S$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ podgraf 
+\begin_inset Formula $G\Leftrightarrow\exists S\subseteq VG,F\subseteq EG\ni:H=\left(G-F\right)-S$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $uv=e\in EG$
+\end_inset
+
+.
+ 
+\begin_inset Formula $G/e$
+\end_inset
+
+ je skrčitev.
+ (identificiramo 
+\begin_inset Formula $u=v$
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ minor 
+\begin_inset Formula $G$
+\end_inset
+
+: 
+\begin_inset Formula $H=f_{1}...f_{k}G$
+\end_inset
+
+ za 
+\begin_inset Formula $f_{i}$
+\end_inset
+
+ skrčitev/odstranjevanje
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $VG^{+}e\coloneqq VG\cup\left\{ x_{e}\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $EG^{+}e\coloneqq EG\setminus e\cup\left\{ x_{e}u,x_{e}v\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $VG^{+}e$
+\end_inset
+
+ je subdivizija, 
+\begin_inset Formula $e\in EG$
+\end_inset
+
+.
+ Na 
+\begin_inset Formula $e$
+\end_inset
+
+ dodamo vozlišče.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ subdivizija 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow H=G^{+}\left\{ e_{1},\dots,e_{k}\right\} ^{+}\left\{ f_{1}\dots f_{j}\right\} ^{+}\dots$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Stopnja vozlišč se s subdivizijo ne poveča.
+\end_layout
+
+\begin_layout Standard
+Glajenje 
+\begin_inset Formula $G^{-}v$
+\end_inset
+
+, 
+\begin_inset Formula $v\in VG$
+\end_inset
+
+ je obrat subdivizije.
+ 
+\begin_inset Formula $\deg_{G}v=2$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $H$
+\end_inset
+
+ sta homeomorfna, če sta subdivizija istega grafa.
+\end_layout
+
+\begin_layout Standard
+Kartezični produkt: 
+\begin_inset Formula $V\left(G\square H\right)\coloneqq VG\times VH$
+\end_inset
+
+, 
+\begin_inset Formula $E\left(G\square H\right)\coloneqq\left\{ \left\{ \left(g,h\right),\left(g',h'\right)\right\} ;g=g'\wedge hh'\in EH\vee h=h'\wedge gg'\in EG\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(\mathcal{G},\square\right)$
+\end_inset
+
+ monoid, enota 
+\begin_inset Formula $K_{1}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $Q_{n}\cong Q_{n-1}\square K_{2}=Q_{n-2}\square K_{2}^{\square,2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Disjunktna unija: 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $H$
+\end_inset
+
+ disjunktna.
+ 
+\begin_inset Formula $V\left(G\cup H\right)\coloneqq VG\cup VH$
+\end_inset
+
+, 
+\begin_inset Formula $E\left(G\cup H\right)\coloneqq EG\cup EH$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G,H$
+\end_inset
+
+ dvodelna 
+\begin_inset Formula $\Rightarrow G\square H$
+\end_inset
+
+ dvodelen
+\end_layout
+
+\begin_layout Paragraph
+\begin_inset Formula $k-$
+\end_inset
+
+povezan graf
+\end_layout
+
+\begin_layout Standard
+ima 
+\begin_inset Formula $\geq k+1$
+\end_inset
+
+ vozlišč in ne vsebuje prerezne množice moči 
+\begin_inset Formula $<k$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $X\subseteq VG$
+\end_inset
+
+ je prerezna množica 
+\begin_inset Formula $\Leftrightarrow\Omega\left(G-X\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Y\subseteq EG$
+\end_inset
+
+ prerezna množica povezav 
+\begin_inset Formula $\Leftrightarrow\Omega\left(G-Y\right)>\Omega G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Povezanost grafa: 
+\begin_inset Formula $\kappa G=\max k$
+\end_inset
+
+, da je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan.
+ Primeri: 
+\begin_inset Formula $\kappa K_{n}=n-1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa P_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa C_{n}=2$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa K_{n,m}=\min\left\{ n,m\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $\kappa Q_{n}=n$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa G\text{\ensuremath{\leq}}\delta G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek (Menger): 
+\begin_inset Formula $\left|VG\right|\geq k+1$
+\end_inset
+
+: 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG,uv\not\in EG:\exists k$
+\end_inset
+
+ notranje disjunktnih 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti
+\end_layout
+
+\begin_layout Standard
+Graf je 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan po povezavah, če ne vsebuje prerezne množice povezav moči 
+\begin_inset Formula $<k$
+\end_inset
+
+.
+ Povezanost grafa po povezavah: 
+\begin_inset Formula $\kappa'G=\max k$
+\end_inset
+
+, da je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan po povezavah.
+ Primeri: 
+\begin_inset Formula $\kappa'K_{n}=n-1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa'P_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa'C_{n}=2$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek (Menger'): 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists k$
+\end_inset
+
+ po povezavah disjunktnih 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall G\in\mathcal{G}:\kappa G\leq\kappa'G\leq\delta G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Drevo
+\end_layout
+
+\begin_layout Standard
+je povezan gozd.
+ Gozd je graf brez ciklov.
+\end_layout
+
+\begin_layout Standard
+Drevo z vsaj dvema vozliščema premore dva lista.
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+hspace*{
+\backslash
+fill}
+\end_layout
+
+\end_inset
+
+NTSE:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ drevo 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists!u,v-$
+\end_inset
+
+pot 
+\begin_inset Formula $\Leftrightarrow\Omega G=1\wedge\forall e\in EG:e$
+\end_inset
+
+ most 
+\begin_inset Formula $\Leftrightarrow\Omega G=1\wedge\left|EG\right|=\left|VG\right|-1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vpeto drevo grafa je vpet podgraf, ki je drevo.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\tau G\sim$
+\end_inset
+
+ število vpetih dreves.
+ 
+\begin_inset Formula $\Omega G=1\Leftrightarrow\tau G\geq1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall e\in EG:\tau G=\tau\left(G-e\right)+\tau\left(G/e\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+\text{\text{\text{Laplaceova matrika: }}}L\left(G\right)_{ij}=\begin{cases}
+\deg_{G}v_{i} & ;i=j\\
+-\left(\text{št. uv povezav}\right) & ;\text{drugače}
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek (Kirchoff): za 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan multigraf je 
+\begin_inset Formula $\forall i:\tau G=\det\left(LG\text{ brez \ensuremath{i}te vrstice in \ensuremath{i}tega stolpca}\right)$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\tau K_{n}=n^{n-2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Prüferjeva koda, če lahko linearno uredimo vozlišča: Ponavljaj, dokler 
+\begin_inset Formula $VG\not=\emptyset$
+\end_inset
+
+: vzemi prvi list, ga odstrani in v vektor dodaj njegovega soseda.
+\end_layout
+
+\begin_layout Standard
+Blok je maksimalen povezan podgraf brez prereznih vozlišč.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\tau G=\tau B_{1}\cdot\cdots\cdot\tau B_{k}$
+\end_inset
+
+ za bloke 
+\begin_inset Formula $\vec{B}$
+\end_inset
+
+ grafa 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Eulerjev
+\end_layout
+
+\begin_layout Standard
+sprehod v m.grafu vsebuje vse povezave po enkrat.
+\end_layout
+
+\begin_layout Standard
+Eulerjev obhod je sklenjen eulerjev sprehod.
+\end_layout
+
+\begin_layout Standard
+Eulerjev graf premore eulerjev obhod.
+\end_layout
+
+\begin_layout Standard
+Za povezan multigraf 
+\begin_inset Formula $G$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Leftrightarrow\forall v\in VG:\deg_{G}v$
+\end_inset
+
+ sod
+\end_layout
+
+\begin_layout Standard
+Fleuryjev algoritem za eulerjev obhod v eulerjevem grafu: Začnemo v poljubni
+ povezavi, jo izbrišemo, nadaljujemo na mostu le, če nimamo druge možne
+ povezave.
+\end_layout
+
+\begin_layout Standard
+Dekompozicija: delitev na povezavno disjunktne podgrafe.
+\end_layout
+
+\begin_layout Standard
+Dekompozicija je lepa, če so podgrafi izomorfni.
+ (
+\begin_inset Formula $\exists$
+\end_inset
+
+ za 
+\begin_inset Formula $P_{5,2}$
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Standard
+Vsak eulerjev graf premore dekompozicijo v cikle.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Q_{n}$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Leftrightarrow n$
+\end_inset
+
+ sod
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{m,n,p}$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Leftrightarrow m,n,p$
+\end_inset
+
+ iste parnosti
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\wedge H$
+\end_inset
+
+ eulerjev 
+\begin_inset Formula $\Rightarrow G\square H$
+\end_inset
+
+ eulerjev
+\end_layout
+
+\begin_layout Paragraph
+Hamiltonov
+\end_layout
+
+\begin_layout Standard
+cikel vsebuje vsa vozlišča grafa.
+\end_layout
+
+\begin_layout Standard
+Hamilton graf premore Hamiltonov cikel.
+\end_layout
+
+\begin_layout Standard
+Hamiltonova pot vsebuje vsa vozlišča.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ hamiltonov, 
+\begin_inset Formula $S\subseteq VG\Rightarrow\Omega\left(G-S\right)\leq\left|S\right|$
+\end_inset
+
+ torej:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\exists S\in VG:\Omega\left(G-S\right)>\left|S\right|\Rightarrow G$
+\end_inset
+
+ ni hamiltonov.
+ Primer: 
+\begin_inset Formula $G$
+\end_inset
+
+ vsebuje prerezno vozlišče 
+\begin_inset Formula $\Rightarrow G$
+\end_inset
+
+ ni hamiltonov.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{n,m}$
+\end_inset
+
+ je hamiltonov 
+\begin_inset Formula $\Leftrightarrow m=n$
+\end_inset
+
+ (za 
+\begin_inset Formula $S$
+\end_inset
+
+ vzamemo 
+\begin_inset Formula $\min\left\{ m,n\right\} $
+\end_inset
+
+)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left|VG\right|\geq3,\forall u,v\in VG:\deg_{G}u+\deg_{G}v\geq\left|VG\right|\Rightarrow G$
+\end_inset
+
+ hamil.
+\end_layout
+
+\begin_layout Standard
+Dirac: 
+\begin_inset Formula $\left|VG\right|\geq3,\forall u\in VG:\deg_{G}u\geq\frac{\left|VG\right|}{2}\Rightarrow G$
+\end_inset
+
+ hamilton.
+\end_layout
+
+\begin_layout Paragraph
+Ravninski
+\end_layout
+
+\begin_layout Standard
+graf brez sekanja povezav narišemo v ravnino
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{2,3}$
+\end_inset
+
+ je ravninski, 
+\begin_inset Formula $K_{3,3}$
+\end_inset
+
+, 
+\begin_inset Formula $K_{5}$
+\end_inset
+
+, 
+\begin_inset Formula $C_{5}\square C_{5}$
+\end_inset
+
+ in 
+\begin_inset Formula $P_{5,2}$
+\end_inset
+
+ niso ravninski.
+\end_layout
+
+\begin_layout Standard
+Vložitev je ravninski graf z ustrezno risbo v ravnini.
+\end_layout
+
+\begin_layout Standard
+Lica so sklenjena območja vložitve brez vozlišč in povezav.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ lahko vložimo v ravnino 
+\begin_inset Formula $\Leftrightarrow G$
+\end_inset
+
+ lahko vložimo na sfero.
+\end_layout
+
+\begin_layout Standard
+Dolžina lica 
+\begin_inset Formula $F\sim lF$
+\end_inset
+
+ je št.
+ povezav obhoda lica.
+\end_layout
+
+\begin_layout Standard
+Drevo je ravninski graf z enim licem dolžine 
+\begin_inset Formula $2\left|ET\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\sum_{F\in FG}lF=2\left|EG\right|$
+\end_inset
+
+, 
+\begin_inset Formula $lF\geq gG$
+\end_inset
+
+ za ravninski 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $2\left|EG\right|=\sum_{F\in FG}lF\geq\sum_{F\in FG}gG=gG\left|FG\right|$
+\end_inset
+
+ (
+\begin_inset Formula $G$
+\end_inset
+
+ ravn.)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski 
+\begin_inset Formula $\Rightarrow\left|EG\right|\geq\frac{gG}{2}$
+\end_inset
+
+
+\begin_inset Formula $\left|FG\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Euler: 
+\begin_inset Formula $\left|VG\right|-\left|EG\right|+\left|FG\right|=1+\Omega G$
+\end_inset
+
+ za ravninski 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski, 
+\begin_inset Formula $\left|VG\right|\geq3\Rightarrow\left|EG\right|\leq3\left|VG\right|-6$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski brez 
+\begin_inset Formula $C_{3}$
+\end_inset
+
+, 
+\begin_inset Formula $\left|VG\right|\geq3\Rightarrow\left|EG\right|\text{\ensuremath{\leq2\left|VG\right|-4}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Triangulacija je taka vložitev, da so vsa lica omejena s 
+\begin_inset Formula $C_{3}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+V maksimalen ravninski graf ne moremo dodati povezave, da bi ostal ravninski.
+ 
+\begin_inset Formula $\sim$
+\end_inset
+
+ Ni pravi vpet podgraf ravn.
+ grafa.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{5}-e$
+\end_inset
+
+ je maksimalen ravninski graf 
+\begin_inset Formula $\forall e\in EK_{5}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski.
+ NTSE: 
+\begin_inset Formula $G$
+\end_inset
+
+ triangulacija 
+\begin_inset Formula $\Leftrightarrow G$
+\end_inset
+
+ maksimalen ravninski 
+\begin_inset Formula $\Leftrightarrow\left|EG\right|=3\left|VG\right|-6$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ vsaka subdivizija 
+\begin_inset Formula $G$
+\end_inset
+
+ ravninska.
+\end_layout
+
+\begin_layout Standard
+Kuratovski: 
+\begin_inset Formula $G$
+\end_inset
+
+ ravn.
+ 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ ne vsebuje subdivizije 
+\begin_inset Formula $K_{5}$
+\end_inset
+
+ ali 
+\begin_inset Formula $K_{3,3}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Wagner: 
+\begin_inset Formula $G$
+\end_inset
+
+ ravn.
+ 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ niti 
+\begin_inset Formula $K_{5}$
+\end_inset
+
+ niti 
+\begin_inset Formula $K_{3,3}$
+\end_inset
+
+ nista njegova minorja
+\end_layout
+
+\begin_layout Standard
+Zunanje-ravninski ima vsa vozlišča na robu zunanjega lica.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $2-$
+\end_inset
+
+povezan zunanje-ravninski je 
+\series bold
+hamiltonov
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+Zunanje-ravninski 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $\left|VG\right|\geq2$
+\end_inset
+
+ ima vozlišče stopnje 
+\begin_inset Formula $\leq2$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Barvanje vozlišč
+\end_layout
+
+\begin_layout Standard
+je taka 
+\begin_inset Formula $C:VG\to\left\{ 1..k\right\} \Leftrightarrow\forall uv\in EG:Cu\not=Cv$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Kromatično število 
+\begin_inset Formula $\chi G$
+\end_inset
+
+ je najmanjši 
+\begin_inset Formula $k$
+\end_inset
+
+, 
+\begin_inset Formula $\ni:\exists$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+barvanje 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\chi C_{n}=\begin{cases}
+2 & ;n\text{ sod}\\
+3 & ;n\text{ lih}
+\end{cases}$
+\end_inset
+
+, 
+\begin_inset Formula $\chi G\leq\left|VG\right|,\chi G=\left|VG\right|\Leftrightarrow G\cong K_{n}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ podgraf 
+\begin_inset Formula $G\Rightarrow\chi G\geq\chi H$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\chi\text{bipartitni}=2$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Barvni razred so vsa vozlišča iste barve.
+ Je brez povezav 
+\begin_inset Formula $\sim$
+\end_inset
+
+ 
+\series bold
+neodvisna množica
+\series default
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\chi G=k\Leftrightarrow\chi G\leq k\wedge\chi G\geq k$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Klično št.: 
+\begin_inset Formula $\omega G\coloneqq$
+\end_inset
+
+ 
+\begin_inset Formula $\left|V\right|$
+\end_inset
+
+ najv.
+ polnega podgr.
+ v 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\omega G\leq\chi G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Trditev: 
+\begin_inset Formula $\forall n\in\mathbb{N}\exists G\in\mathcal{G}\ni:\omega G=2\wedge\chi G=n$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Požrešno barvanje: Vozlišča v poljubnem vrstnem redu barvamo z najnižno
+ možno barvo.
+\end_layout
+
+\begin_layout Standard
+Vedno 
+\begin_inset Formula $\exists$
+\end_inset
+
+ vrstni red, ki vrne barvanje s 
+\begin_inset Formula $\chi G$
+\end_inset
+
+ barvami.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall G\in\mathcal{G}:\chi G\leq\Delta G+1$
+\end_inset
+
+, če 
+\begin_inset Formula $G$
+\end_inset
+
+ ni regularen celo 
+\begin_inset Formula $\chi G\leq\Delta G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Brooks: 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan, 
+\begin_inset Formula $G$
+\end_inset
+
+ ni poln niti lihi cikel 
+\begin_inset Formula $\Rightarrow\chi G\leq\Delta G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+let 
+\begin_inset Formula $d_{1}\geq\cdots\geq d_{n}$
+\end_inset
+
+ stopnje.
+ 
+\begin_inset Formula $\chi G=1+\max_{i=1}^{n}\min\left\{ d_{i},i-1\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Spoj 
+\begin_inset Formula $G\oplus H$
+\end_inset
+
+ je disjunktna unija 
+\begin_inset Formula $G$
+\end_inset
+
+,
+\begin_inset Formula $H$
+\end_inset
+
+ z vsemi pov.
+ med njima
+\end_layout
+
+\begin_layout Standard
+Disjunktna unija je unija brez preseka.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\chi\left(G\oplus H\right)=\chi G+\chi H$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Ravninski graf ima 
+\begin_inset Formula $\chi G\leq4$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Barvanje povezav
+\end_layout
+
+\begin_layout Standard
+Povezavi s skupnim vozl.
+ 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ razl.
+ barvi.
+\end_layout
+
+\begin_layout Standard
+Kromatični indeks 
+\begin_inset Formula $\chi'G$
+\end_inset
+
+ je najm.
+ št.
+ barv za barv.
+ pov.
+ 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\chi'C_{n}=\begin{cases}
+2 & ;n\text{ sod}\\
+3 & ;n\text{ lih}
+\end{cases}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vizing: 
+\begin_inset Formula $\forall G\in\mathcal{G}:\Delta G\geq\chi'G\geq\Delta G+1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\chi G\in\left\{ \Delta G\text{ razred I.},\Delta G+1\text{ razred II.}\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $K_{2k+1}\in$
+\end_inset
+
+ II., 
+\begin_inset Formula $K_{2k}\in$
+\end_inset
+
+ I., bipartitni 
+\begin_inset Formula $\in$
+\end_inset
+
+ I.
+\end_layout
+
+\begin_layout Paragraph
+Neodvisne množice
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $I\subseteq VG$
+\end_inset
+
+ je neodvisna, če je podgraf, induciran z 
+\begin_inset Formula $I$
+\end_inset
+
+, brez povezav.
+\end_layout
+
+\begin_layout Standard
+Neodvisnostno št 
+\begin_inset Formula $\alpha G$
+\end_inset
+
+ je moč največje neodvisne množice.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\alpha K_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha C_{n}=\lfloor\frac{n}{2}\rfloor$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\alpha G\leq\sum_{i=1}^{k}\alpha H_{i}$
+\end_inset
+
+, kjer so 
+\begin_inset Formula $H_{i}$
+\end_inset
+
+ disjunktni inducirani podgr.
+ 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall G\in\mathcal{G}:\alpha G\cdot\chi G\geq\left|VG\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\frac{\left|VG\right|}{\chi G}\leq\alpha G\leq\left|VG\right|-\frac{\left|EG\right|}{\Delta G}$
+\end_inset
+
+ — zgornja in spodnja meja.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Iu$
+\end_inset
+
+ je 
+\begin_inset Formula $\alpha$
+\end_inset
+
+ poddrevesa s korenom 
+\begin_inset Formula $u$
+\end_inset
+
+ v 
+\begin_inset Formula $T$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\alpha T=Ir=\max\left\{ 1+\sum_{u\text{ sinovi }r}Iu,\sum_{u\text{ vnuki }r}Iu\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Dominantne množice
+\end_layout
+
+\begin_layout Standard
+Neodvisna 
+\begin_inset Formula $I\subseteq G$
+\end_inset
+
+ je maksimalna 
+\begin_inset Formula $\sim\nexists S\text{ neodvisna}\subseteq G\ni:I\subset S\sim$
+\end_inset
+
+ ni prava 
+\begin_inset Formula $\subset$
+\end_inset
+
+ neodv.
+ mn.
+\end_layout
+
+\begin_layout Standard
+Dominantna množica je maksimalna neodvisna, kjer ima vsako vozlišče iz 
+\begin_inset Formula $G\setminus I$
+\end_inset
+
+ soseda v 
+\begin_inset Formula $I$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $D\subseteq VG$
+\end_inset
+
+ dominira 
+\begin_inset Formula $X\subseteq VG$
+\end_inset
+
+, če je vsako vozlišče iz 
+\begin_inset Formula $X$
+\end_inset
+
+ v 
+\begin_inset Formula $D$
+\end_inset
+
+ ali pa ima soseda v 
+\begin_inset Formula $D$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $N_{G}\left(u\right)=$
+\end_inset
+
+sosedje 
+\begin_inset Formula $u$
+\end_inset
+
+ v 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $N_{G}\left[u\right]=N_{G}\left(u\right)\cup\left\{ u\right\} $
+\end_inset
+
+ zap.
+ sos.
+ 
+\begin_inset Formula $u$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $N_{G}\left[D\right]=\bigcup_{u\in D}N_{G}\left[u\right]$
+\end_inset
+
+ zaprta soseščina množice 
+\begin_inset Formula $D$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $D$
+\end_inset
+
+ dominira 
+\begin_inset Formula $X\sim X\subseteq N_{G}\left[D\right]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $D$
+\end_inset
+
+ dominira 
+\begin_inset Formula $VG\sim D$
+\end_inset
+
+ dominantna množica 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dominacijsko število 
+\begin_inset Formula $\gamma G=$
+\end_inset
+
+ moč največje dom.
+ mn.
+\end_layout
+
+\begin_layout Standard
+Vsaka maksimalna neodvisna mn.
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ je dom.
+ mn.
+ 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\gamma K_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\gamma C_{n}=\lceil\frac{n}{3}\rceil$
+\end_inset
+
+, 
+\begin_inset Formula $\gamma P=3$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za vsak graf brez izoliranih vozlišč 
+\begin_inset Formula $\lceil\frac{\left|VG\right|}{\Delta G+1}\rceil\leq\gamma G\leq\lfloor\frac{\left|VG\right|}{2}\rfloor$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $X$
+\end_inset
+
+ je 2-pakiranje 
+\begin_inset Formula $G$
+\end_inset
+
+, če 
+\begin_inset Formula $\forall x,y\in X:x\not=y\Rightarrow d_{G}\left(x,y\right)\geq3$
+\end_inset
+
+ zdb 
+\begin_inset Formula $\forall x,y\in X:x\not=y\Rightarrow N_{G}x\cap N_{G}y=\emptyset$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Moč največjega 2-pakiranja je 2-pakirno število 
+\begin_inset Formula $\rho G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\Rightarrow\gamma G\geq\rho G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dominantna 
+\begin_inset Formula $X$
+\end_inset
+
+ je popolna koda 
+\begin_inset Formula $G\Leftrightarrow\bigcup_{u\in X}N_{G}\left[u\right]=VG$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Če graf premore popolno kodo 
+\begin_inset Formula $\Rightarrow\gamma G=\rho G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ vpet podgraf 
+\begin_inset Formula $G\Rightarrow\gamma G\leq\gamma H$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Povezan 
+\begin_inset Formula $G$
+\end_inset
+
+ premore vpeto drevo 
+\begin_inset Formula $T\ni:\gamma G=\gamma T$
+\end_inset
+
+.
+ Dokaz: odstrani vse povezave, kjer 
+\begin_inset Formula $G-e$
+\end_inset
+
+ ohrani 
+\begin_inset Formula $\gamma$
+\end_inset
+
+ in povezanost.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left|VG\right|\geq1\Rightarrow\gamma G\leq\chi\overline{G}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Povezana dom.
+ mn.
+ inducira povez.
+ podgraf.
+ — 
+\begin_inset Formula $\gamma_{c}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Neodvisna dom.
+ mn.
+ inducira podgraf brez povezav — 
+\begin_inset Formula $\gamma_{i}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $X$
+\end_inset
+
+ je celotna dom.
+ mn.
+ 
+\begin_inset Formula $\Leftrightarrow\forall v\in VG:N_{G}\left(v\right)\cap X\not=\emptyset$
+\end_inset
+
+ — 
+\begin_inset Formula $\gamma_{t}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G$
+\end_inset
+
+ brez izol.
+ vozl.: 
+\begin_inset Formula $\gamma G\leq\gamma_{t}G\leq2\gamma G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Algebrske strukture z eno operacijo
+\end_layout
+
+\begin_layout Standard
+Grupoid: notranjost, polgrupa: asociativen grupoid, monoid: polgrupa z enoto
+\end_layout
+
+\begin_layout Standard
+Grupa: 
+\begin_inset Formula $\forall a\in A\exists a^{-1}\in A\ni:a\cdot a^{-1}=a^{-1}\cdot a=e$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Potence v monoidu: 
+\begin_inset Formula $n\in\mathbb{N}_{0}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $a^{0}=e$
+\end_inset
+
+, 
+\begin_inset Formula $a^{n}=a\cdot a^{n-1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(a\cdot b\right)^{-1}=b^{-1}\cdot a^{-1}$
+\end_inset
+
+, 
+\begin_inset Formula $a^{n}\cdot a^{m}=a^{n+m},\left(a^{n}\right)^{m}=a^{nm}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(a^{-1}\right)^{n}=\left(a^{n}\right)^{-1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(\mathbb{Z}_{m},+_{m}\right)$
+\end_inset
+
+ grupa, 
+\begin_inset Formula $\left(\mathbb{Z}_{m},\cdot_{n}\right)$
+\end_inset
+
+ monoid
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $k\in\mathbb{Z}_{n}$
+\end_inset
+
+ obrnljiv v 
+\begin_inset Formula $\left(\mathbb{Z}_{n},\cdot_{n}\right)\Leftrightarrow k\perp n\sim\gcd\left\{ k,n\right\} =1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Komutativni grupi pravimo abelova.
+\end_layout
+
+\begin_layout Standard
+Cayleyeva tabela je predpis za grupoid 
+\begin_inset Formula $\cdot:A^{2}\to A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+red 
+\begin_inset Formula $a$
+\end_inset
+
+ je najmanjši 
+\begin_inset Formula $n\in\mathbb{N}\ni:a^{n}=e$
+\end_inset
+
+ oz.
+ 
+\begin_inset Formula $\infty$
+\end_inset
+
+, če ne obstaja.
+\end_layout
+
+\begin_layout Standard
+Def.: 
+\begin_inset Formula $H\subseteq G$
+\end_inset
+
+ podgrupa, če je grupa za isto operacijo.
+\end_layout
+
+\begin_layout Standard
+Izrek: 
+\begin_inset Formula $H\subseteq G$
+\end_inset
+
+ podgrupa 
+\begin_inset Formula $\Leftrightarrow\forall x,y\in H:x^{-1}y\in H$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Trivialni podgrupi 
+\begin_inset Formula $G$
+\end_inset
+
+ sta 
+\begin_inset Formula $\left\{ e\right\} $
+\end_inset
+
+ in 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ 
+\begin_inset space \hfill{}
+\end_inset
+
+Ciklična podgrupa:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left\langle a\right\rangle \coloneqq\left\{ a^{n};\forall n\in\mathbb{Z}\right\} $
+\end_inset
+
+ je podgrupa v 
+\begin_inset Formula $G$
+\end_inset
+
+ za 
+\begin_inset Formula $a\in\text{grupa }G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Center grupe: 
+\begin_inset Formula $ZG=\left\{ a\in G;\forall x\in G:ax=xa\right\} $
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(ZG,\cdot\right)$
+\end_inset
+
+je podgrupa grupe 
+\begin_inset Formula $\left(G,\cdot\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Permutacijske grupe
+\end_layout
+
+\begin_layout Standard
+Permutacija je bijekcija 
+\begin_inset Formula $A\to A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+perm.
+ gr.
+ so permutacije 
+\begin_inset Formula $A$
+\end_inset
+
+, ki tvorijo grupo za 
+\begin_inset Formula $\circ$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Polna simetrična grupa 
+\begin_inset Formula $S_{n}\coloneqq\left\{ \pi:\left\{ 1..n\right\} \to\left\{ 1..n\right\} ;\pi\text{ bijekcija}\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Alternirajoča grupa 
+\begin_inset Formula $A_{n}\coloneqq\left\{ \pi\in S_{n};\pi\text{ soda}\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Permutacija kot produkt disjunktnih ciklov dolžin 
+\begin_inset Formula $l_{1},\dots,l_{n}$
+\end_inset
+
+ je soda, če je 
+\begin_inset Formula $\left(l_{1}-1\right)+\cdots+\left(l_{n}-1\right)$
+\end_inset
+
+ sod.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left|S_{n}\right|=n!$
+\end_inset
+
+, 
+\begin_inset Formula $\left|A_{n}\right|=\frac{n!}{2}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(G,\circ\right)$
+\end_inset
+
+,
+\begin_inset Formula $\left(H,*\right)$
+\end_inset
+
+ grupi.
+ 
+\begin_inset Formula $f:G\to H$
+\end_inset
+
+ je hm
+\begin_inset Formula $\varphi\Leftrightarrow\forall x,y\in G:f\left(x\circ y\right)=fx*fy$
+\end_inset
+
+.
+ 
+\begin_inset Formula $f$
+\end_inset
+
+ je še celo im
+\begin_inset Formula $\varphi\Leftrightarrow f$
+\end_inset
+
+ bijekcija.
+\end_layout
+
+\begin_layout Standard
+Grupi sta izomorfni 
+\begin_inset Formula $\sim\text{\ensuremath{\exists}}$
+\end_inset
+
+im
+\begin_inset Formula $\varphi$
+\end_inset
+
+ med njima: 
+\begin_inset Formula $G\approx H$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vsaka grupa je izomorfna neki permutacijski grupi
+\end_layout
+
+\begin_layout Paragraph
+Odseki in podgrupe edinke
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(G,\circ\right)$
+\end_inset
+
+ grupa, 
+\begin_inset Formula $H\subseteq G$
+\end_inset
+
+.
+ za 
+\begin_inset Formula $a\in G$
+\end_inset
+
+: 
+\begin_inset Formula $aH=\left\{ ah;h\in H\right\} ,Ha=\left\{ ha;h\in H\right\} $
+\end_inset
+
+.
+ Za 
+\begin_inset Formula $H$
+\end_inset
+
+ podgrupo pravimo 
+\begin_inset Formula $aH$
+\end_inset
+
+ levi odsek 
+\begin_inset Formula $H$
+\end_inset
+
+ in 
+\begin_inset Formula $Ha$
+\end_inset
+
+ desni.
+ Velja: 
+\begin_inset Formula $a\in aH$
+\end_inset
+
+, 
+\begin_inset Formula $aH=H\Leftrightarrow a\in H$
+\end_inset
+
+, 
+\begin_inset Formula $aH=bH\nabla aH\cap bH=\emptyset$
+\end_inset
+
+, 
+\begin_inset Formula $aH=bH\Leftrightarrow a^{-1}b\in H$
+\end_inset
+
+, 
+\begin_inset Formula $\left|aH\right|=\left|bH\right|$
+\end_inset
+
+, 
+\begin_inset Formula $aH=Ha\Leftrightarrow H=aHa^{-1}$
+\end_inset
+
+, 
+\begin_inset Formula $aH\subseteq G\Leftrightarrow a\in H$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+general linear 
+\begin_inset Formula $GL_{2}\mathbb{R}=\left\{ A\in M_{2,2}\mathbb{R},\det A\not=0\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+special linear 
+\begin_inset Formula $SL_{2}\mathbb{R}=\left\{ A\in M_{2,2}\mathbb{R},\det A=1\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Lagrange: 
+\begin_inset Formula $H$
+\end_inset
+
+ podgrupa končne grupe 
+\begin_inset Formula $G\Rightarrow\left|H\right|\text{ deli }\left|G\right|$
+\end_inset
+
+ in število različnih levih/desnih odeskov po 
+\begin_inset Formula $H$
+\end_inset
+
+ je 
+\begin_inset Formula $\frac{\left|G\right|}{\left|H\right|}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $a\in$
+\end_inset
+
+ končne grupe 
+\begin_inset Formula $G\Rightarrow$
+\end_inset
+
+ red 
+\begin_inset Formula $a$
+\end_inset
+
+ deli 
+\begin_inset Formula $\left|G\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Podgrupe edinke in faktorske grupe
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H$
+\end_inset
+
+ podgrupa 
+\begin_inset Formula $G$
+\end_inset
+
+ je edinka 
+\begin_inset Formula $\Leftrightarrow\forall a\in G:aH=Ha\sim aHa^{-1}=H$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Podgrupa konjugiranka 
+\begin_inset Formula $H^{a}\coloneqq aHa^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $H\triangleleft G\sim H$
+\end_inset
+
+ edinka v 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow\forall a\in G:H^{A}=H$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+V Abelovi grupi je vsaka podgrupa edinka.
+\end_layout
+
+\begin_layout Standard
+Center je podgrupa edinka.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $G/H\coloneqq\left\{ aH:a\in G\right\} $
+\end_inset
+
+ z oper.
+ 
+\begin_inset Formula $\left(aH\right)*\left(bH\right)=\left(ab\right)H$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek o faktorskih grupah: 
+\begin_inset Formula $H\triangleleft G\Rightarrow\left(G/H,*\right)$
+\end_inset
+
+ grupa
+\end_layout
+
+\begin_layout Paragraph
+Kolobarji in polja
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(R,+,\cdot\right)\text{kolobar}\Rightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $\left(R,+\right)$
+\end_inset
+
+ abelova grupa, 
+\begin_inset Formula $\left(R,\cdot\right)$
+\end_inset
+
+ polgrupa, distributivnost.
+\end_layout
+
+\begin_layout Standard
+Kolobar je komutativen, če je 
+\begin_inset Formula $\cdot$
+\end_inset
+
+ komutativna.
+\end_layout
+
+\begin_layout Standard
+Kolobar je z enoto, če obstaja enota za 
+\begin_inset Formula $\cdot$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Direktna vsota kolobarjev 
+\begin_inset Formula $\left(R\oplus S,+,\cdot\right)$
+\end_inset
+
+ je kolobar.
+ 
+\begin_inset Formula $R\oplus S=R\times S$
+\end_inset
+
+, 
+\begin_inset Formula $+$
+\end_inset
+
+ in 
+\begin_inset Formula $\cdot$
+\end_inset
+
+ po komponentah.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $R$
+\end_inset
+
+ in 
+\begin_inset Formula $S$
+\end_inset
+
+ komutativna 
+\begin_inset Formula $\Rightarrow R\oplus S$
+\end_inset
+
+ komutativen.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $R$
+\end_inset
+
+ in 
+\begin_inset Formula $S$
+\end_inset
+
+ z enoto 
+\begin_inset Formula $\Rightarrow R\oplus S$
+\end_inset
+
+ z enoto.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $R$
+\end_inset
+
+ kolobar.
+ 
+\begin_inset Formula $S\subseteq R$
+\end_inset
+
+ je za podedovani operaciji kolobar, če je 
+\begin_inset Formula $\left(S,+,\cdot\right)$
+\end_inset
+
+ kolobar.
+ Izrek: 
+\begin_inset Formula $S$
+\end_inset
+
+ podkolobar 
+\begin_inset Formula $\Leftrightarrow0\in S$
+\end_inset
+
+ in zaprta za 
+\begin_inset Formula $-,\cdot$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Center kolobarja: 
+\begin_inset Formula $ZR=\left\{ a\in R;\forall x\in R:ax=xa\right\} $
+\end_inset
+
+ je podk.
+\end_layout
+
+\begin_layout Paragraph
+Delitelji niča in celi kolobarji
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(\mathbb{Z}_{n},+_{n},\cdot_{n}\right)$
+\end_inset
+
+ je vedno kol.
+\end_layout
+
+\begin_layout Standard
+Def.: Če v kolobarju velja 
+\begin_inset Formula $ab=0$
+\end_inset
+
+ in 
+\begin_inset Formula $a\not=0$
+\end_inset
+
+ in 
+\begin_inset Formula $b\not=0$
+\end_inset
+
+, sta 
+\begin_inset Formula $a$
+\end_inset
+
+ in 
+\begin_inset Formula $b$
+\end_inset
+
+ delitelja niča.
+\end_layout
+
+\begin_layout Standard
+Pravilo krajšanja: 
+\begin_inset Formula $\forall a,b,c\in R,a\not=0:ab=ac\Rightarrow b=c$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $R$
+\end_inset
+
+ cel 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ komutativen z enoto 
+\begin_inset Formula $1\not=0$
+\end_inset
+
+ brez deliteljev niča.
+\end_layout
+
+\begin_layout Standard
+Za komutativen z enoto 
+\begin_inset Formula $1\not=0$
+\end_inset
+
+ velja: cel 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ velja prav.
+ krajš.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(R,+,\cdot\right)$
+\end_inset
+
+ z enoto 
+\begin_inset Formula $1\not=0$
+\end_inset
+
+ je polje, če je 
+\begin_inset Formula $\left(R\setminus\left\{ 0\right\} ,\cdot\right)$
+\end_inset
+
+ abelova g.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(R,+,\cdot\right)$
+\end_inset
+
+ z enoto 
+\begin_inset Formula $1\not=0$
+\end_inset
+
+ je obseg, če je 
+\begin_inset Formula $\left(R\setminus\left\{ 0\right\} ,\cdot\right)$
+\end_inset
+
+ grupa.
+\end_layout
+
+\begin_layout Standard
+Vsako polje je cel kolobar.
+\end_layout
+
+\begin_layout Standard
+Polje je cel kolobar z multiplik.
+ inverzi za vse neničelne.
+\end_layout
+
+\begin_layout Standard
+Končen cel kolobar 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ polje.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall n\in\mathbb{N}\setminus\left\{ 1\right\} :\mathbb{Z}_{n}$
+\end_inset
+
+ cel 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $\mathbb{Z}_{n}$
+\end_inset
+
+ polje 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $n$
+\end_inset
+
+ praštevilo
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $S\subseteq$
+\end_inset
+
+ polje 
+\begin_inset Formula $R$
+\end_inset
+
+ je podpolje za poded.
+ operaciji, če je 
+\begin_inset Formula $S$
+\end_inset
+
+ polje.
+ Zadošča preveriti 
+\begin_inset Formula $1\in S$
+\end_inset
+
+, zaprtost za 
+\begin_inset Formula $-$
+\end_inset
+
+ in deljenje z nenič.ž
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $n\cdot'a$
+\end_inset
+
+ naj pomeni 
+\begin_inset Formula $a+\cdots+a$
+\end_inset
+
+ 
+\begin_inset Formula $n$
+\end_inset
+
+-krat za 
+\begin_inset Formula $n\in\mathbb{N}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Karakteristika kolobarja je najmanjši 
+\begin_inset Formula $n\in N\ni:\forall a\in R:n\cdot'a=0$
+\end_inset
+
+.
+ Če ne obstaja, pravimo char
+\begin_inset Formula $R=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Izrek: Če je red1
+\begin_inset Formula $=n\Rightarrow$
+\end_inset
+
+ char
+\begin_inset Formula $R=n$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izrek: Za cel kolobar 
+\begin_inset Formula $R$
+\end_inset
+
+ je char
+\begin_inset Formula $R=0$
+\end_inset
+
+ ali char
+\begin_inset Formula $R=$
+\end_inset
+
+ praštevilo
+\end_layout
+
+\begin_layout Paragraph
+Ideali
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $S$
+\end_inset
+
+ podkolobar 
+\begin_inset Formula $R$
+\end_inset
+
+ je ideal 
+\begin_inset Formula $R\Leftrightarrow\forall r\in R,s\in S:rs,sr\in S$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Primer: večkratniki 
+\begin_inset Formula $n$
+\end_inset
+
+ so za fiksen 
+\begin_inset Formula $n$
+\end_inset
+
+ ideal v 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $I\subseteq R$
+\end_inset
+
+ je ideal 
+\begin_inset Formula $\Leftrightarrow0\in I$
+\end_inset
+
+, zaprt za 
+\begin_inset Formula $-$
+\end_inset
+
+, zaprt za zunanje množ.
+\end_layout
+
+\begin_layout Standard
+Operaciji nad ideali: 
+\begin_inset Formula $I\overset{+}{\cdot}J=\left\{ i\overset{+}{\cdot}j;\forall i\in I,j\in J\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za ideala 
+\begin_inset Formula $I,J$
+\end_inset
+
+ v 
+\begin_inset Formula $R$
+\end_inset
+
+ sta 
+\begin_inset Formula $I+J$
+\end_inset
+
+ in 
+\begin_inset Formula $I\cdot J$
+\end_inset
+
+ zopet ideala v 
+\begin_inset Formula $R$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+V aditivne odseke 
+\begin_inset Formula $R/I=\left\{ a+I;\forall a\in R\right\} $
+\end_inset
+
+ vpeljemo operaciji 
+\begin_inset Formula $\left(a+I\right)\overset{+}{\cdot}\left(b+I\right)=\left(a\overset{+}{\cdot}b\right)I$
+\end_inset
+
+.
+ Če je 
+\begin_inset Formula $I$
+\end_inset
+
+ ideal v 
+\begin_inset Formula $R$
+\end_inset
+
+, je 
+\begin_inset Formula $R/I$
+\end_inset
+
+ za operaciji kolobar.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/ds2/teor.lyx b/šola/ds2/teor.lyx
new file mode 100644
index 0000000..2d03ffc
--- /dev/null
+++ b/šola/ds2/teor.lyx
@@ -0,0 +1,7560 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{hyperref}
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\usepackage{amsmath}
+\usepackage{tikz}
+\newcommand{\udensdash}[1]{%
+    \tikz[baseline=(todotted.base)]{
+        \node[inner sep=1pt,outer sep=0pt] (todotted) {#1};
+        \draw[densely dashed] (todotted.south west) -- (todotted.south east);
+    }%
+}%
+\DeclareMathOperator{\Lin}{Lin}
+\DeclareMathOperator{\rang}{rang}
+\DeclareMathOperator{\sled}{sled}
+\DeclareMathOperator{\Aut}{Aut}
+\DeclareMathOperator{\red}{red}
+\DeclareMathOperator{\karakteristika}{char}
+\usepackage{algorithm,algpseudocode}
+\providecommand{\corollaryname}{Posledica}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+theorems-ams
+\end_modules
+\maintain_unincluded_children false
+\language slovene
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 2cm
+\topmargin 2cm
+\rightmargin 2cm
+\bottommargin 2cm
+\headheight 2cm
+\headsep 2cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Odgovori na vprašanja za teoretični del izpita DS2 IŠRM 2023/24
+\end_layout
+
+\begin_layout Author
+
+\noun on
+Anton Luka Šijanec
+\end_layout
+
+\begin_layout Date
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+today
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Abstract
+Vprašanja je zbral profesor Sandi Klavžar.
+ Odgovore sem sestavil po svojih zapiskih z njegovih predavanj in drugih
+ virih.
+\end_layout
+
+\begin_layout Standard
+\begin_inset CommandInset toc
+LatexCommand tableofcontents
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Section
+Slog
+\end_layout
+
+\begin_layout Itemize
+Z 
+\begin_inset Formula $M_{m,n}\mathbb{F}$
+\end_inset
+
+ označim množico matrik z 
+\begin_inset Formula $m$
+\end_inset
+
+ vrsticami in 
+\begin_inset Formula $n$
+\end_inset
+
+ stolpci nad poljem 
+\begin_inset Formula $\mathbb{F}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Znak za množenje ali dvojiški operator v grupi izpuščam.
+ V bigrupoidih izpuščen operator pomeni množenje (drugo operacijo).
+\end_layout
+
+\begin_layout Section
+Vprašanja in odgovori
+\end_layout
+
+\begin_layout Subsection
+Kaj je stopnja vozlišča grafa in kaj pravi lema o rokovanju? Kako dokažemo
+ to lemo?
+\end_layout
+
+\begin_layout Standard
+Naj bo 
+\begin_inset Formula $G$
+\end_inset
+
+ graf.
+\end_layout
+
+\begin_layout Definition*
+Stopnja vozlišča grafa 
+\begin_inset Formula $\deg v$
+\end_inset
+
+ za 
+\begin_inset Formula $v\in VG$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Ko eniški operator zahteva en operand, izpustim oklepaje.
+ Primer: 
+\begin_inset Formula $\sin x$
+\end_inset
+
+, 
+\begin_inset Formula $VG$
+\end_inset
+
+, 
+\begin_inset Formula $\chi G$
+\end_inset
+
+, 
+\begin_inset Formula $M_{2,2}\mathbb{R}$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+ predstavlja število povezav, ki imajo to vozlišče kot krajišče.
+ 
+\begin_inset Formula $\deg v=\left|\left\{ e\in EG;v\in e\right\} \right|$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Povezava v grafu je množica natanko dveh vozlišč, toda oklepaje in vejico
+ izpuščam.
+ Za 
+\begin_inset Formula $u,v\in VG$
+\end_inset
+
+ pišem 
+\begin_inset Formula $uv\in EG$
+\end_inset
+
+.
+ Na 
+\begin_inset Formula $e\in EG$
+\end_inset
+
+ je torej veljavno gledati kot na množico, zato je 
+\begin_inset Formula $v\in e$
+\end_inset
+
+ smiseln izraz.
+\end_layout
+
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Zapis množic: 
+\begin_inset Formula $\left\{ e\in A;Pe\right\} $
+\end_inset
+
+ pomeni vse take elemente 
+\begin_inset Formula $A$
+\end_inset
+
+, za katere velja predikat 
+\begin_inset Formula $P$
+\end_inset
+
+.
+ Ta predikat je običajno izraz.
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Incidenčna matrika 
+\begin_inset Formula $BG$
+\end_inset
+
+ za 
+\begin_inset Formula $VG=\left\{ v_{1},\dots,v_{n}\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $EG=\left\{ e_{1},\dots,e_{n}\right\} $
+\end_inset
+
+ je široka 
+\begin_inset Formula $m$
+\end_inset
+
+ in visoka 
+\begin_inset Formula $n$
+\end_inset
+
+ in velja 
+\begin_inset Formula $BG_{ij}=\begin{cases}
+1 & ;v_{i}\in e_{i}\\
+0 & ;\text{sicer}
+\end{cases}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Lemma*
+Lema o rokovanju pravi, da je dvakratnik števila povezav enak vsoti vseh
+ stopenj vozlišč v grafu
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Ko je omenjen graf, je običajno mišljen neusmerjen končen graf.
+\end_layout
+
+\end_inset
+
+, torej
+\begin_inset Formula 
+\[
+\sum_{v\in VG}\deg v=2\left|EG\right|.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Remark*
+Posledica leme o rokovanju je, da je v vsakem grafu sodo vozlišč lihe stopnje,
+ saj je vsota soda.
+\end_layout
+
+\begin_layout Proof
+Če po vozliščih preštevamo povezave, ki se stikajo vozlišča, bi vsako povezavo
+ šteli dvakrat; vsakič za eno njeno krajišče.
+ ZDB V vsakem stolpcu incidenčne matrike sta natanko dve enici.
+ Torej je enic 
+\begin_inset Formula $2\left|EG\right|$
+\end_inset
+
+.
+ V vsaki vrstici incidenčne matrike je toliko enic, kolikor je stopnja 
+\begin_inset Formula $i-$
+\end_inset
+
+tega vozlišča, torej je enic 
+\begin_inset Formula $\sum_{v\in VG}\deg v$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Pojasnite sprehod, sklenjen sprehod, pot v grafu, cikel v grafu.
+ Pokažite, da vsak graf, ki vsebuje sklenjen sprehod lihe dolžine, vsebuje
+ tudi cikel lihe dolžine.
+\end_layout
+
+\begin_layout Standard
+Naj bo 
+\begin_inset Formula $G=\left(VG,EG\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Sprehod v 
+\begin_inset Formula $G$
+\end_inset
+
+ je zaporedje vozlišč 
+\begin_inset Formula $v_{0},\dots,v_{k}$
+\end_inset
+
+, 
+\begin_inset Formula $k\geq0$
+\end_inset
+
+, tako da je 
+\begin_inset Formula $\forall i\in\left\{ 1..k\right\} :v_{i}v_{i+1}\in EG$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+S 
+\begin_inset Formula $\left\{ k..n\right\} $
+\end_inset
+
+ označujem množico celih števil od vključno 
+\begin_inset Formula $k$
+\end_inset
+
+ do vključno 
+\begin_inset Formula $n$
+\end_inset
+
+, kot to naredi 
+\family typewriter
+bash.
+ 
+\family default
+Nekateri bi 
+\begin_inset Formula $\left\{ 1..n\right\} $
+\end_inset
+
+ označili z 
+\begin_inset Formula $\left[n\right]$
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+.
+ Dolžina sprehoda je število prehojenih povezav.
+ Sprehod je 
+\series bold
+sklenjen
+\series default
+, če 
+\begin_inset Formula $v_{0}=v_{k}$
+\end_inset
+
+.
+ Sprehod je 
+\series bold
+enostaven
+\series default
+, če so vsa vozlišča medsebojno različna, z izjemo 
+\begin_inset Formula $v_{0}$
+\end_inset
+
+ sme biti 
+\begin_inset Formula $v_{k}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+Če v grafu 
+\begin_inset Formula $\exists$
+\end_inset
+
+ sprehod med dvema vozliščema, med njima obstaja tudi enostaven sprehod.
+\end_layout
+
+\begin_layout Proof
+Naj bo 
+\begin_inset Formula $Q$
+\end_inset
+
+ sprehod med 
+\begin_inset Formula $u$
+\end_inset
+
+ in 
+\begin_inset Formula $v$
+\end_inset
+
+.
+ Če je enostaven, je najden, sicer se ponovita vsaj dve vozlišči in je 
+\begin_inset Formula $Q$
+\end_inset
+
+ oblike 
+\begin_inset Formula $u,\dots,x,\dots,x,\dots,v$
+\end_inset
+
+.
+ Sprehodu priredimo 
+\begin_inset Formula $Q'$
+\end_inset
+
+, ki odstrani pot od prvega do zadnjega podvojenega vozlišča 
+\begin_inset Formula $x$
+\end_inset
+
+, torej je 
+\begin_inset Formula $Q'$
+\end_inset
+
+ oblike 
+\begin_inset Formula $u,\dots,x,\dots,v$
+\end_inset
+
+ (pravzaprav smo odstranili cikel iz poti).
+ Če je 
+\begin_inset Formula $Q'$
+\end_inset
+
+ enostaven, je iskani, sicer postopek ponovimo in v končno korakih se postopek
+ ustavi, saj na vsakem koraku za vsaj 2 zmanjšamo dolžino sicer končno dolgega
+ sprehoda.
+\end_layout
+
+\begin_layout Definition*
+Pot v grafu je podgraf, ki je enostaven sprehod med dvema vozliščema in
+ je graf Pot (
+\begin_inset Formula $P_{n}$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Remark*
+Ko govorimo o različnosti/enakosti poti, mislimo različnost/enakost poti
+ kot podgraf.
+ Poti sta torej enaki natanko tedaj, ko sta zaporedji (ne množici) vozlišč
+ poti enaki.
+\end_layout
+
+\begin_layout Definition*
+Cikel v grafu je podgraf, ki je enostaven sklenjen sprehod dolžine vsaj
+ 3.
+\end_layout
+
+\begin_layout Claim*
+Če med dvema vozliščema v grafu 
+\begin_inset Formula $\exists$
+\end_inset
+
+ dve različni poti, potem graf premore cikel.
+\end_layout
+
+\begin_layout Proof
+Naj bosta 
+\begin_inset Formula $P$
+\end_inset
+
+ in 
+\begin_inset Formula $P'$
+\end_inset
+
+ dve različni poti od 
+\begin_inset Formula $u$
+\end_inset
+
+ do 
+\begin_inset Formula $v$
+\end_inset
+
+.
+ Naj bo 
+\begin_inset Formula $x$
+\end_inset
+
+ zadnje (če gledamo usmerjeno 
+\begin_inset Formula $u,v-$
+\end_inset
+
+pot) vozlišče, ki je skupno 
+\begin_inset Formula $P$
+\end_inset
+
+ in 
+\begin_inset Formula $P'$
+\end_inset
+
+.
+ 
+\begin_inset Formula $x$
+\end_inset
+
+ je lahko 
+\begin_inset Formula $u$
+\end_inset
+
+.
+ Naj bo 
+\begin_inset Formula $g$
+\end_inset
+
+ prvo naslednje vozlišče za 
+\begin_inset Formula $x$
+\end_inset
+
+, ki je na 
+\begin_inset Formula $P$
+\end_inset
+
+ in 
+\begin_inset Formula $P'$
+\end_inset
+
+.
+ Unija podpoti 
+\begin_inset Formula $P$
+\end_inset
+
+ od 
+\begin_inset Formula $x$
+\end_inset
+
+ do 
+\begin_inset Formula $y$
+\end_inset
+
+ in podpoti 
+\begin_inset Formula $P'$
+\end_inset
+
+ od 
+\begin_inset Formula $x$
+\end_inset
+
+ do 
+\begin_inset Formula $y$
+\end_inset
+
+ določa iskani cikel v grafu.
+\end_layout
+
+\begin_layout Claim*
+Če graf premore sklenjen sprehod lihe dolžine, potem premore cikel lihe
+ dolžine.
+\end_layout
+
+\begin_layout Proof
+Indukcija po dolžini sprehoda.
+ Naj bo 
+\begin_inset Formula $m$
+\end_inset
+
+ dolžina sprehoda.
+\end_layout
+
+\begin_layout Proof
+Baza: 
+\begin_inset Formula $m=3$
+\end_inset
+
+, najmanjši sklenjen lihi sprehod je cikel dolžine 3, 
+\begin_inset Formula $m=4$
+\end_inset
+
+, drugi najmanjši sklenjen lihi sprehod je cikel dolžine 4.
+\end_layout
+
+\begin_layout Proof
+Korak: Naj bo 
+\begin_inset Formula $Q$
+\end_inset
+
+ poljuben sklenjen sprehod dolžine 
+\begin_inset Formula $m\geq5$
+\end_inset
+
+.
+ Če je 
+\begin_inset Formula $Q$
+\end_inset
+
+ enostaven, je cikel po definiciji, sicer se vsaj eno vozlišče na sprehodu
+ vsaj ponovi: 
+\begin_inset Formula $u,x_{1},\dots,x_{i-1},x_{i},x_{i+1},\dots,x_{j-1},x_{j},x_{j+1},\dots,v$
+\end_inset
+
+ in velja 
+\begin_inset Formula $x_{i}=x_{j}$
+\end_inset
+
+.
+ Poglejmo sprehoda 
+\begin_inset Formula $Q'=x_{i},\dots,x_{j}=x_{i}$
+\end_inset
+
+ in 
+\begin_inset Formula $Q''=u,x_{1},\dots,x_{i},x_{j+1},\dots,x_{m}=u$
+\end_inset
+
+.
+ 
+\begin_inset Formula $Q'$
+\end_inset
+
+ in 
+\begin_inset Formula $Q''$
+\end_inset
+
+ sta sklenjena sprehoda z dolžinama 
+\begin_inset Formula $m'$
+\end_inset
+
+ in 
+\begin_inset Formula $m''$
+\end_inset
+
+ in velja 
+\begin_inset Formula $m=m'+m''$
+\end_inset
+
+.
+ Ker je 
+\begin_inset Formula $m$
+\end_inset
+
+ lih, mora biti lih natanko en izmed 
+\begin_inset Formula $m'$
+\end_inset
+
+ in 
+\begin_inset Formula $m''$
+\end_inset
+
+.
+ BSŠ
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Brez Škode za Splošnost trdimo, da
+\end_layout
+
+\end_inset
+
+ 
+\begin_inset Formula $m'$
+\end_inset
+
+ lih in 
+\begin_inset Formula $m'<m$
+\end_inset
+
+, zato po I.
+ P.
+ 
+\begin_inset Formula $m'$
+\end_inset
+
+ vsebuje lih cikel.
+\end_layout
+
+\begin_layout Subsection
+Kaj so dvodelni grafi? Kako jih karakteriziramo? Kako dokažemo to karakterizacij
+o?
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Pomožni glagol biti med osebkom in povedkovnikom izpuščam.
+ 
+\begin_inset Quotes gld
+\end_inset
+
+
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen
+\begin_inset Quotes grd
+\end_inset
+
+ namesto 
+\begin_inset Quotes gld
+\end_inset
+
+
+\begin_inset Formula $G$
+\end_inset
+
+ je dvodelen
+\begin_inset Quotes grd
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow\exists A,B\subseteq VG\ni:A\cup B=VG,A\cap B=\emptyset\ni:\forall uv\in EG:u\in A,v\in B\vee v\in A,u\in B$
+\end_inset
+
+.
+ ZDB
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Z Drugimi Besedami:
+\end_layout
+
+\end_inset
+
+ obstaja razdelitev vozlišč na dve množici, da induciran podgraf posamezne
+ množice ne vsebuje povezav.
+ S 
+\begin_inset Formula $K_{m,n}$
+\end_inset
+
+ označimo poln dvodelni graf 
+\begin_inset Formula $\left|A\right|=m$
+\end_inset
+
+, 
+\begin_inset Formula $\left|B\right|=n$
+\end_inset
+
+.
+ Paru 
+\begin_inset Formula $\left(A,B\right)$
+\end_inset
+
+ pravimo dvodelna razdelitev.
+\end_layout
+
+\begin_layout Theorem*
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ ne vsebuje lihih ciklov.
+\end_layout
+
+\begin_layout Proof
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ vsaka komponenta 
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelna, zato BSŠ 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ Očitno: Če 
+\begin_inset Formula $G$
+\end_inset
+
+ vsebuje lih cikel, zagotovo ni dvodelen, saj ne moremo razdeliti niti množice
+ vozlišč cikla.
+ S skico dokažemo, da sodi cikli so dvodelni, lihi pa niso (narišemo cikel
+ kot pot v obliki skeletne formule nenasičenega acikličnega alkana in povežemo
+ prvo in zadnje vozlišče).
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ je po predpostavki brez lihih ciklov ****.
+ Izberimo poljubno 
+\begin_inset Formula $x_{0}\in VG$
+\end_inset
+
+.
+ Naj bo 
+\begin_inset Formula $A=\left\{ u\in VG;d_{G}\left(u,x_{0}\right)\text{ sod}\right\} ,B=\left\{ u\in VG;d_{G}\left(u,x_{0}\right)\text{ lih}\right\} $
+\end_inset
+
+.
+ 
+\begin_inset Formula $x_{0}$
+\end_inset
+
+ je torej v 
+\begin_inset Formula $A$
+\end_inset
+
+, saj je 
+\begin_inset Formula $d_{G}\left(x_{0},x_{0}\right)=0$
+\end_inset
+
+.
+ Trdimo, da je 
+\begin_inset Formula $\left(A,B\right)$
+\end_inset
+
+ dvodelna razdelitev 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Razdelitev je, ker je 
+\begin_inset Formula $A\cup B=\emptyset$
+\end_inset
+
+ in 
+\begin_inset Formula $A\cup B=VG$
+\end_inset
+
+.
+ Za dvodelnost pa mora veljati 
+\begin_inset Formula $\forall X\in\left\{ A,B\right\} :\forall u,v\in X:uv\not\in EG$
+\end_inset
+
+.
+ Preverimo za splošen fiksen 
+\begin_inset Formula $X\in\left\{ A,B\right\} $
+\end_inset
+
+: Naj bosta 
+\begin_inset Formula $u,v\in X$
+\end_inset
+
+, BSŠ 
+\begin_inset Formula $d_{G}\left(x_{0},u\right)>d_{G}\left(x_{0},v\right)$
+\end_inset
+
+ *.
+ Ločimo dva primera:
+\end_layout
+
+\begin_deeper
+\begin_layout Description
+\begin_inset Formula $d_{G}\left(x_{0},u\right)\not=d_{G}\left(x_{0},v\right):$
+\end_inset
+
+ Vsled iste parnosti velja 
+\begin_inset Formula $\left|d_{G}\left(x_{0},u\right)-d_{G}\left(x_{0},v\right)\right|\geq2$
+\end_inset
+
+ ***.
+ PDDRAA
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Pa Denimo Da sledeča trditev ne drži (Reductio Ad Absurdum)
+\end_layout
+
+\end_inset
+
+ 
+\begin_inset Formula $uv\in EG$
+\end_inset
+
+, tedaj se 
+\begin_inset Formula $d_{G}\left(x_{0},u\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $d_{G}\left(x_{0},v\right)$
+\end_inset
+
+ razlikujeta za največ 1 in velja 
+\begin_inset Formula $d_{G}\left(x_{0},u\right)\leq d_{G}\left(x_{0},v\right)+1$
+\end_inset
+
+ **.
+ Iz * in ** sledi 
+\begin_inset Formula $d_{G}\left(x_{0},u\right)-d_{G}\left(x_{0},v\right)=1$
+\end_inset
+
+, kar je v 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+ s trditvijo ***.
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $d_{G}\left(x_{0},u\right)=d_{G}\left(x_{0},v\right):$
+\end_inset
+
+ Naj bo 
+\begin_inset Formula $P_{x}$
+\end_inset
+
+ najkrajša 
+\begin_inset Formula $x_{0},x-$
+\end_inset
+
+pot.
+ PDDRAA 
+\begin_inset Formula $uv\in EG$
+\end_inset
+
+, tedaj 
+\begin_inset Formula $P_{u}=\left\{ ...P_{v},v\right\} $
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Prefiksni razširitveni operator 
+\begin_inset Formula $...$
+\end_inset
+
+ razširi množico oziroma v tem primeru zaporedje v seznam elementov.
+ S tem lahko množico uporabimo kot seznam.
+ Povzemam operator 
+\family typewriter
+...
+
+\family default
+ iz javascripta.
+\end_layout
+
+\end_inset
+
+, torej 
+\begin_inset Formula $\left|P_{u}\right|=1+\left|P_{v}\right|$
+\end_inset
+
+.
+ Cikel, ki ga tvorijo 
+\begin_inset Formula $P_{u}$
+\end_inset
+
+, 
+\begin_inset Formula $P_{v}$
+\end_inset
+
+ in povezava 
+\begin_inset Formula $uv$
+\end_inset
+
+, je torej dolžine 
+\begin_inset Formula $2\left|P_{v}\right|+1$
+\end_inset
+
+, kar je liho število, kar je v 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+ s trditvijo ****.
+\end_layout
+
+\end_deeper
+\begin_layout Subsection
+Kaj je homomorfizem grafov, izomorfizem grafov in avtomorfizem grafa? Kaj
+ je to 
+\begin_inset Formula $\Aut\left(G\right)$
+\end_inset
+
+? Kakšno algebrsko strukturo ima?
+\end_layout
+
+\begin_layout Standard
+Naj bosta 
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $H$
+\end_inset
+
+ grafa.
+\end_layout
+
+\begin_layout Definition*
+Preslikava 
+\begin_inset Formula $f:VG\to VH$
+\end_inset
+
+ je hm
+\begin_inset Formula $\varphi$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+homomorfizem
+\end_layout
+
+\end_inset
+
+ grafov 
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $H\Leftrightarrow\forall u,v\in VG:uv\in EG\Rightarrow fufv\in EH$
+\end_inset
+
+, ZDB če slika povezave v povezave.
+\end_layout
+
+\begin_layout Remark*
+Če je 
+\begin_inset Formula $f$
+\end_inset
+
+ hm
+\begin_inset Formula $\text{\ensuremath{\varphi}},$
+\end_inset
+
+porodi preslikavo 
+\begin_inset Formula $f':EG\to EH$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $f:VK_{n,m}\to VK_{2}$
+\end_inset
+
+ s predpisom 
+\begin_inset Formula $fx=\begin{cases}
+u & ;x\in A\\
+v & ;x\in B
+\end{cases}$
+\end_inset
+
+ je hm
+\begin_inset Formula $\varphi$
+\end_inset
+
+.
+ 
+\begin_inset Formula $K_{2}$
+\end_inset
+
+ je homomorfna slika vsakega dvodelnega grafa.
+\end_layout
+
+\begin_layout Definition*
+Če je hm
+\begin_inset Formula $\varphi$
+\end_inset
+
+ surjektiven po povezavah in vozliščih, je epimorfizem.
+ Če je injektiven na vozliščih (in posledično na povezavah), je monomorfizem
+ ali vložitev.
+ Vložitev je izometrična, če 
+\begin_inset Formula $\forall u,v\in VG:d_{G}\left(u,v\right)=d_{H}\left(fu,fv\right)$
+\end_inset
+
+ ZDB ohranja razdalje.
+\end_layout
+
+\begin_layout Claim*
+Če sta 
+\begin_inset Formula $f:VG\to VH$
+\end_inset
+
+ in 
+\begin_inset Formula $g:VH\to VK$
+\end_inset
+
+ hm
+\begin_inset Formula $\varphi$
+\end_inset
+
+, je 
+\begin_inset Formula $g\circ f:VG\to VK$
+\end_inset
+
+ spet hm
+\begin_inset Formula $\varphi$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Če je 
+\begin_inset Formula $f:VG\to VH$
+\end_inset
+
+ bijekcija, hm
+\begin_inset Formula $\varphi$
+\end_inset
+
+ in 
+\begin_inset Formula $f^{-1}$
+\end_inset
+
+ hm
+\begin_inset Formula $\varphi$
+\end_inset
+
+ (ZDB slika nepovezave v nepovezave), je 
+\begin_inset Formula $f$
+\end_inset
+
+ im
+\begin_inset Formula $\varphi$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+izomorfizem
+\end_layout
+
+\end_inset
+
+ grafov 
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $H$
+\end_inset
+
+.
+ ZDB 
+\begin_inset Formula $f:VG\to VH$
+\end_inset
+
+ im
+\begin_inset Formula $\varphi$
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow f$
+\end_inset
+
+ bijekcija 
+\begin_inset Formula $\wedge\forall u,v\in VG:uv\in EG\Leftrightarrow fufv\in EH$
+\end_inset
+
+.
+ Če 
+\begin_inset Formula $\text{\ensuremath{\exists}}$
+\end_inset
+
+ im
+\begin_inset Formula $\varphi$
+\end_inset
+
+ grafov 
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $H$
+\end_inset
+
+, pravimo, da sta 
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $H$
+\end_inset
+
+ izomorfna in pišemo 
+\begin_inset Formula $G\cong H$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $\cong$
+\end_inset
+
+ je na 
+\begin_inset Formula $\mathcal{G}$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+množica vseh grafov
+\end_layout
+
+\end_inset
+
+ ekvivalenčna (refleksivna, simetrična, tranzitivna).
+\end_layout
+
+\begin_layout Definition*
+im
+\begin_inset Formula $\varphi$
+\end_inset
+
+ 
+\begin_inset Formula $G\to G$
+\end_inset
+
+ je am
+\begin_inset Formula $\varphi$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+avtomorfizem
+\end_layout
+
+\end_inset
+
+.
+ Vse am
+\begin_inset Formula $\varphi$
+\end_inset
+
+ grafa 
+\begin_inset Formula $G$
+\end_inset
+
+ združimo v množico in jo opremimo z operacijo komponiranja.
+ Dobimo 
+\begin_inset Quotes gld
+\end_inset
+
+grupo avtomorfizmov grafa 
+\begin_inset Formula $G$
+\end_inset
+
+
+\begin_inset Quotes grd
+\end_inset
+
+, ki jo označimo z 
+\begin_inset Formula $\Aut G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\Aut C_{4}=\left(\left\{ id,\left(2,4\right),\left(12\right)\left(34\right),\left(1234\right),\left(13\right),\left(13\right)\left(24\right),\left(1423\right),\left(4321\right)\right\} ,\circ\right)$
+\end_inset
+
+, 
+\begin_inset Formula $\Aut K_{n}=\left(S_{n},\circ\right)$
+\end_inset
+
+ za 
+\begin_inset Formula $S_{n}$
+\end_inset
+
+ množica vseh permutacij 
+\begin_inset Formula $n$
+\end_inset
+
+ elementov,
+\begin_inset Formula $\Aut P_{n}=\left(\left\{ id,f\left(i\right)=n-i-1\right\} ,\circ\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Fact*
+Za vsako končno grupo 
+\begin_inset Formula $X$
+\end_inset
+
+ 
+\begin_inset Formula $\exists$
+\end_inset
+
+ graf 
+\begin_inset Formula $G\ni:\Aut G=X$
+\end_inset
+
+.
+ Algebra 
+\begin_inset Formula $\subseteq$
+\end_inset
+
+ Diskretne strukture.
+ Dokaz na magisteriju.
+\end_layout
+
+\begin_layout Subsection
+Kaj pomeni, da je graf 
+\begin_inset Formula $H$
+\end_inset
+
+ minor grafa 
+\begin_inset Formula $G$
+\end_inset
+
+? Kdaj sta dva grafa homeomorfna? Pojasni operacijo kartezičnega produkta
+ grafov.
+\end_layout
+
+\begin_layout Definition*
+
+\series bold
+Odstranjevanje vozlišč
+\series default
+: za neko 
+\begin_inset Formula $S\subseteq VG$
+\end_inset
+
+ je 
+\begin_inset Formula $G-S$
+\end_inset
+
+ graf, ki ga dobimo, ko iz 
+\begin_inset Formula $G$
+\end_inset
+
+ odstranimo vozlišča in vozliščem pripadajoče povezave iz 
+\begin_inset Formula $S$
+\end_inset
+
+.
+ Za 
+\begin_inset Formula $S=\left\{ u\right\} $
+\end_inset
+
+ pišemo tudi 
+\begin_inset Formula $G-u$
+\end_inset
+
+.
+ 
+\series bold
+Odstranjevanje povezav
+\series default
+: za neko 
+\begin_inset Formula $F\subseteq EG$
+\end_inset
+
+ je 
+\begin_inset Formula $G-F$
+\end_inset
+
+ graf, ki ga dobimo, ko iz 
+\begin_inset Formula $G$
+\end_inset
+
+ odstranimo povezave iz 
+\begin_inset Formula $F$
+\end_inset
+
+.
+ Za 
+\begin_inset Formula $S=\left\{ e\right\} $
+\end_inset
+
+ pišemo tudi 
+\begin_inset Formula $G-e$
+\end_inset
+
+.
+ 
+\series bold
+Skrčitev povezave
+\series default
+: za 
+\begin_inset Formula $e=\left\{ u,v\right\} \in EG$
+\end_inset
+
+ je 
+\begin_inset Formula $G/e$
+\end_inset
+
+ graf, ki ga dobimo tako, da identificiramo 
+\begin_inset Formula $u$
+\end_inset
+
+ in 
+\begin_inset Formula $v$
+\end_inset
+
+ in odstranimo morebitne vzporedne povezave, s čimer odstranimo tudi zanko
+ 
+\begin_inset Formula $\left\{ u,u=v\right\} $
+\end_inset
+
+.
+ 
+\begin_inset Formula $G/e_{1}/e_{2}/\dots/e_{n}$
+\end_inset
+
+ označimo z 
+\begin_inset Formula $G/\left\{ e_{1},e_{2},\dots,e_{n}\right\} $
+\end_inset
+
+ .
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $H$
+\end_inset
+
+ minor 
+\begin_inset Formula $G\Leftrightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $H$
+\end_inset
+
+ dobimo iz 
+\begin_inset Formula $G$
+\end_inset
+
+ z nekim zaporedjem operacij, kjer so dovoljene operacije odstranjevanje
+ vozlišča, odstranjevanje povezave in skrčitev povezave.
+ Ekvivalentno je 
+\begin_inset Formula $H$
+\end_inset
+
+ minor 
+\begin_inset Formula $G$
+\end_inset
+
+, če ga lahko dobimo iz nekega podgrafa 
+\begin_inset Formula $G$
+\end_inset
+
+, ki mu skrčimo poljubno povezav.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Subdivizija povezav: 
+\begin_inset Formula $G\to G^{+}e$
+\end_inset
+
+ za 
+\begin_inset Formula $e\in EG$
+\end_inset
+
+: 
+\begin_inset Formula $VG^{+}e=VG\cup\left\{ x_{e}\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $EG^{+}e=EG\setminus e\cup\left\{ x_{e}u,x_{e}v\right\} $
+\end_inset
+
+.
+ Graf 
+\begin_inset Formula $H$
+\end_inset
+
+ je subdivizija 
+\begin_inset Formula $G\Leftrightarrow H$
+\end_inset
+
+ dobimo iz 
+\begin_inset Formula $G$
+\end_inset
+
+ z zaporedjem subdivizij povezav 
+\begin_inset Formula $G$
+\end_inset
+
+ ZDB povezave v 
+\begin_inset Formula $G$
+\end_inset
+
+ zamenjamo s poljubno dolčimi potmi.
+ Relacija je refleksivna (
+\begin_inset Formula $G$
+\end_inset
+
+ subdivizija 
+\begin_inset Formula $G$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Glajenje vozlišča 
+\begin_inset Formula $u$
+\end_inset
+
+ stopnje 
+\begin_inset Formula $2$
+\end_inset
+
+: (obratna operacija od subdivizije) 
+\begin_inset Formula $G^{-}u=\left(VG\setminus u,\left(EG\setminus\left\{ e,f\right\} \right)\cup\left\{ \left\{ v,w\right\} \right\} \right)$
+\end_inset
+
+, kjer sta 
+\begin_inset Formula $e$
+\end_inset
+
+ in 
+\begin_inset Formula $f$
+\end_inset
+
+ povezavi, ki vsebujeta 
+\begin_inset Formula $u$
+\end_inset
+
+, 
+\begin_inset Formula $v$
+\end_inset
+
+ in 
+\begin_inset Formula $w$
+\end_inset
+
+ pa njuni drugi krajišči (tisti krajišči, ki nista 
+\begin_inset Formula $u$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Grafa sta homeomorfna, če sta izomorfna po gladitvi vseh vozlišč stopnje
+ 2.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Naj bosta 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $H$
+\end_inset
+
+ grafa.
+ Kartezični produkt grafov 
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $H$
+\end_inset
+
+ označimo z 
+\begin_inset Formula $G\square H$
+\end_inset
+
+: 
+\begin_inset Formula $V\left(G\square H\right)=VG\times VH,E\left(G\square H\right)=\left\{ \left(g,h\right)\left(g',h'\right);g=g'\wedge hh'\in EH\vee h=h'\wedge gg'\in EG\right\} $
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Remark*
+\begin_inset Formula $\left(\mathcal{G},\square\right)$
+\end_inset
+
+ je monoid, kajti 
+\begin_inset Formula $K_{1}$
+\end_inset
+
+ je enota, operacija je komutativna in notranja.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $K_{2}\square K_{2}\cong C_{4}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Kaj so to prerezna vozlišča in prerezne povezave grafa? Kdaj je graf 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan in kaj je to povezanost grafa?
+\end_layout
+
+\begin_layout Standard
+Naj bo 
+\begin_inset Formula $G$
+\end_inset
+
+ graf z 
+\begin_inset Formula $m$
+\end_inset
+
+ vozlišči.
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $u,v\in VG$
+\end_inset
+
+ sta v isti povezani komponenti, če v 
+\begin_inset Formula $G$
+\end_inset
+
+ obstaja sprehod med njima.
+ Število komponent 
+\begin_inset Formula $G$
+\end_inset
+
+ označimo z 
+\begin_inset Formula $\Omega G$
+\end_inset
+
+.
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\Leftrightarrow\Omega G=1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Remark*
+Biti v isti komponenti je ekvivalenčna relacija (refleksivna, simetrična
+ in tranzitivna).
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $x\in VG$
+\end_inset
+
+ prerezno 
+\begin_inset Formula $\Leftrightarrow\Omega\left(G-x\right)>\Omega G$
+\end_inset
+
+.
+ 
+\begin_inset Formula $e\in EG$
+\end_inset
+
+ prerezna 
+\begin_inset Formula $\sim e$
+\end_inset
+
+ most 
+\begin_inset Formula $\Leftrightarrow\Omega\left(G-e\right)>\Omega G$
+\end_inset
+
+.
+ 
+\begin_inset Formula $X\subseteq VG$
+\end_inset
+
+ je prerezna množica 
+\begin_inset Formula $\Leftrightarrow\Omega\left(G-X\right)>\Omega G$
+\end_inset
+
+.
+ 
+\begin_inset Formula $X\subseteq EG$
+\end_inset
+
+ je prerezna množica povezav 
+\begin_inset Formula $\Leftrightarrow\Omega\left(G-X\right)>\Omega G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Povezan graf 
+\begin_inset Formula $G$
+\end_inset
+
+ je 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan 
+\begin_inset Formula $\Leftrightarrow\left|VG\right|\geq k+1\wedge\forall X\text{ prerezna}\subseteq VG:\left|X\right|>=k$
+\end_inset
+
+ prerezna.
+ ZDB ima vsaj 
+\begin_inset Formula $k+1$
+\end_inset
+
+ vozlišč in 
+\series bold
+ne
+\series default
+ vsebuje prerezne množice moči manjše od 
+\begin_inset Formula $k$
+\end_inset
+
+.
+ Povezanost grafa 
+\begin_inset Formula $G\sim\kappa G$
+\end_inset
+
+ je največji 
+\begin_inset Formula $k\in\mathbb{N}$
+\end_inset
+
+, za katerega je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan ZDB najmanjše število vozlišč, ki jih moramo odstraniti iz grafa,
+ da graf ne bo več povezan.
+\end_layout
+
+\begin_layout Remark*
+\begin_inset Formula $\kappa G=k\Rightarrow G$
+\end_inset
+
+ nima prerezne množice moči 
+\begin_inset Formula $<k$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\kappa K_{n}=n-1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa P_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa C_{n}=2$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa K_{m,n}=\min\left\{ n,m\right\} $
+\end_inset
+
+, 
+\begin_inset Formula $\kappa Q_{n}=n$
+\end_inset
+
+, 
+\begin_inset Formula $\kappa G\leq\delta G$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+\begin_inset Formula $\delta G$
+\end_inset
+
+ je minimalna stopnja vozlišča v grafu 
+\begin_inset Formula $G$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Pojasnite Whitney-ev izrek, ki katekterizira 
+\begin_inset Formula $2-$
+\end_inset
+
+povezane grafe.
+ Skicirajte dokaz tega izreka.
+ Zapišite Mengerjev izrek.
+\end_layout
+
+\begin_layout Definition*
+Poti 
+\begin_inset Formula $\left[p_{1},p_{2},\dots,p_{n-1},p\right]$
+\end_inset
+
+ in 
+\begin_inset Formula $\left[r_{1},r_{2},\dots,r_{m-1},r_{m}\right]$
+\end_inset
+
+ sta notranje disjunktni, če sta disjunktni množici njunih vozlišč izvzemši
+ prvo in zadnje vozlišče.
+\end_layout
+
+\begin_layout Theorem*
+Graf 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $\left|VG\right|\geq2$
+\end_inset
+
+, je 
+\begin_inset Formula $2-$
+\end_inset
+
+povezan 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists$
+\end_inset
+
+ notranje disjunktni 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti.
+\end_layout
+
+\begin_layout Proof
+Dokazujemo ekvivalenco:
+\end_layout
+
+\begin_deeper
+\begin_layout Description
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ Po predpostavki za poljubna 
+\begin_inset Formula $u,v$
+\end_inset
+
+ obstajata dve notranje disjunktni 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti.
+ Dokazujemo, da je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $2-$
+\end_inset
+
+povezan 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ nima prereznega vozlišča 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ ne obstaja prerezna množica, ki je singleton
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+množica moči ena
+\end_layout
+
+\end_inset
+
+.
+ Dokažimo, da poljubno 
+\begin_inset Formula $x\in VG$
+\end_inset
+
+ ni prerezno, torej da po odstranitvi tega vozlišča še vedno obstaja povezava
+ med poljubnima 
+\begin_inset Formula $u,v\in V\left(G-x\right)$
+\end_inset
+
+.
+ Ločimo 3 primere: 
+\begin_inset Formula $x$
+\end_inset
+
+ je na prvi 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti, 
+\begin_inset Formula $x$
+\end_inset
+
+ je na drugi 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti, 
+\begin_inset Formula $x$
+\end_inset
+
+ ni na nobeni izmed 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti.
+ V vsakem primeru sta 
+\begin_inset Formula $u,v$
+\end_inset
+
+ v 
+\begin_inset Formula $G-x$
+\end_inset
+
+ še vedno v isti povezani komponenti.
+ 
+\end_layout
+
+\begin_layout Description
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ Po predpostavki je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $2-$
+\end_inset
+
+povezan.
+ Vzemimo poljubna 
+\begin_inset Formula $u,v\in VG$
+\end_inset
+
+.
+ Indukcija po 
+\begin_inset Formula $d_{G}\left(u,v\right)$
+\end_inset
+
+:
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Baza: 
+\begin_inset Formula $d_{G}\left(u,v\right)=1$
+\end_inset
+
+ (sosednji vozlišči) 
+\begin_inset Formula $G-e$
+\end_inset
+
+ je povezan, sicer je RAAPDD 
+\begin_inset Formula $uv$
+\end_inset
+
+ most, kar bi bilo v 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+ s predpostavko, ker bi bil en izmed 
+\begin_inset Formula $u,v$
+\end_inset
+
+ prerezno, saj ima 
+\begin_inset Formula $G$
+\end_inset
+
+ vsaj 3 vozlišča in ima vsaj eden izmed 
+\begin_inset Formula $u,v$
+\end_inset
+
+ še enega soseda.
+ Sedaj vemo, da 
+\begin_inset Formula $\exists u,v-$
+\end_inset
+
+pot, ki ni 
+\begin_inset Formula $uv$
+\end_inset
+
+.
+ Imamo torej dve notranje disjunktni 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti: 
+\begin_inset Formula $e$
+\end_inset
+
+ in tisto drugo.
+\end_layout
+
+\begin_layout Standard
+Korak: 
+\begin_inset Formula $d_{G}\left(u,v\right)=k\geq2$
+\end_inset
+
+.
+ Naj bo 
+\begin_inset Formula $P$
+\end_inset
+
+ najkrajša 
+\begin_inset Formula $u,v-$
+\end_inset
+
+pot.
+ Predzadnje vozlišče na njej (tik pred 
+\begin_inset Formula $v$
+\end_inset
+
+) naj bo 
+\begin_inset Formula $w$
+\end_inset
+
+.
+ 
+\begin_inset Formula $d_{G}\left(u,w\right)=k-1$
+\end_inset
+
+ in po I.
+ P.
+ obstajata dve notranje disjunktni 
+\begin_inset Formula $u,w-$
+\end_inset
+
+poti 
+\begin_inset Formula $R$
+\end_inset
+
+ in 
+\begin_inset Formula $S$
+\end_inset
+
+.
+ Ločimo primera:
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $v\in VR\cup VS$
+\end_inset
+
+ Tedaj je 
+\begin_inset Formula $v$
+\end_inset
+
+ na ciklu, ki ga tvorita poti 
+\begin_inset Formula $R$
+\end_inset
+
+ in 
+\begin_inset Formula $S$
+\end_inset
+
+ (je na eni izmed poti).
+ Zato obstajata dve notranje disjunktni 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti; ena v eno smer po ciklu, druga v drugo.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $v\not\in VR\cup VS$
+\end_inset
+
+ Tedaj 
+\begin_inset Formula $v$
+\end_inset
+
+ ni na tem ciklu, vendar je sosed 
+\begin_inset Formula $w$
+\end_inset
+
+, ki je na ciklu.
+ 
+\begin_inset Formula $G-w$
+\end_inset
+
+ mora biti povezan, saj smo odstranili le eno vozlišče, torej 
+\begin_inset Formula $\exists u,v-$
+\end_inset
+
+pot 
+\begin_inset Formula $T$
+\end_inset
+
+.
+ Ločimo primera:
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $T\cap\left(VS\cup VR\right)=\left\{ u\right\} $
+\end_inset
+
+ Našli smo dve notranje disjunktni poti v 
+\begin_inset Formula $G$
+\end_inset
+
+: 
+\begin_inset Formula $T$
+\end_inset
+
+ in 
+\begin_inset Formula $\left[\dots R,v\right]$
+\end_inset
+
+ (
+\begin_inset Formula $R$
+\end_inset
+
+ in 
+\begin_inset Formula $wv$
+\end_inset
+
+ povezava).
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left|T\cap\left(VS\cup VR\right)\right|\geq2$
+\end_inset
+
+ Naj bo 
+\begin_inset Formula $x$
+\end_inset
+
+ zadnje (kjer je 
+\begin_inset Formula $v$
+\end_inset
+
+ konec poti) vozlišče na 
+\begin_inset Formula $T$
+\end_inset
+
+, ki je na 
+\begin_inset Formula $R\cup S$
+\end_inset
+
+.
+ BSŠ 
+\begin_inset Formula $x\in VS$
+\end_inset
+
+.
+ 
+\begin_inset Formula $x\not=w$
+\end_inset
+
+ po konstrukciji 
+\begin_inset Formula $T$
+\end_inset
+
+.
+ Dve poti: po 
+\begin_inset Formula $S$
+\end_inset
+
+ do 
+\begin_inset Formula $x$
+\end_inset
+
+ in nato po 
+\begin_inset Formula $T$
+\end_inset
+
+ do 
+\begin_inset Formula $v$
+\end_inset
+
+ ter 
+\begin_inset Formula $\left[\dots R,v\right]$
+\end_inset
+
+ (
+\begin_inset Formula $R$
+\end_inset
+
+ in 
+\begin_inset Formula $wv$
+\end_inset
+
+ povezava).
+\end_layout
+
+\end_deeper
+\end_deeper
+\end_deeper
+\begin_layout Theorem*
+Menger (posplošitev Whitneya): naj bo 
+\begin_inset Formula $G$
+\end_inset
+
+ graf z vsak 
+\begin_inset Formula $k+1$
+\end_inset
+
+ vozlišči.
+ Tedaj je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists k$
+\end_inset
+
+ paroma notranje disjunktnih 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti.
+\end_layout
+
+\begin_layout Definition*
+Graf je 
+\begin_inset Formula $k-$
+\end_inset
+
+ povezan po povezavah, če 
+\begin_inset Formula $\nexists$
+\end_inset
+
+ prerezna množica povezav moči 
+\begin_inset Formula $<k$
+\end_inset
+
+.
+ Povezanost grafa 
+\begin_inset Formula $G$
+\end_inset
+
+ po povezavah (
+\begin_inset Formula $\kappa'G$
+\end_inset
+
+) je največji 
+\begin_inset Formula $k$
+\end_inset
+
+, da je 
+\begin_inset Formula $G$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan po povezavah.
+\end_layout
+
+\begin_layout Theorem*
+Menger': 
+\begin_inset Formula $G$
+\end_inset
+
+ je 
+\begin_inset Formula $k-$
+\end_inset
+
+povezan po povezavah 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in VG\exists k$
+\end_inset
+
+ po povezavah notranje disjunktnih 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Theorem*
+\begin_inset Formula $\forall G\in\mathcal{G}:\kappa G\leq\kappa'G$
+\end_inset
+
+ in 
+\begin_inset Formula $\kappa'G\leq\delta G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Dokaze zadnjih treh izrekov najdete na magisteriju.
+\end_layout
+
+\begin_layout Subsection
+Kaj je drevo in kaj je gozd? Katere katekterizacije dreves poznate?
+\end_layout
+
+\begin_layout Definition*
+Gozd je graf brez ciklov.
+ Drevo je povezan gozd.
+ List je vozlišče stopnje 1.
+\end_layout
+
+\begin_layout Lemma
+\begin_inset CommandInset label
+LatexCommand label
+name "lem:Vsako-drevo-z"
+
+\end_inset
+
+Vsako drevo z vsaj dvema vozliščema premore list.
+\end_layout
+
+\begin_layout Proof
+Naj bo 
+\begin_inset Formula $T$
+\end_inset
+
+ drevo, 
+\begin_inset Formula $v\in VT$
+\end_inset
+
+ poljubno.
+ 
+\begin_inset Formula $\left|VT\right|\geq2\wedge T$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\Rightarrow v$
+\end_inset
+
+ ima soseda 
+\begin_inset Formula $v_{1}$
+\end_inset
+
+.
+ Če je 
+\begin_inset Formula $v_{1}$
+\end_inset
+
+ edini sosed 
+\begin_inset Formula $v$
+\end_inset
+
+, je 
+\begin_inset Formula $v$
+\end_inset
+
+ list, sicer je tudi 
+\begin_inset Formula $v_{2}\not=v_{1}$
+\end_inset
+
+ sosed 
+\begin_inset Formula $v$
+\end_inset
+
+.
+ Če je 
+\begin_inset Formula $v$
+\end_inset
+
+ edini sosed 
+\begin_inset Formula $v_{2}$
+\end_inset
+
+, je 
+\begin_inset Formula $v_{2}$
+\end_inset
+
+ list, sicer je tudi 
+\begin_inset Formula $v_{3}$
+\end_inset
+
+ sosed 
+\begin_inset Formula $v$
+\end_inset
+
+.
+ Postopek ponavljamo, dokler ne najdemo lista.
+ V grafu ni ciklov in graf je končen, zato se postopek ustavi.
+\end_layout
+
+\begin_layout Lemma
+\begin_inset CommandInset label
+LatexCommand label
+name "lem:-drevo"
+
+\end_inset
+
+
+\begin_inset Formula $T$
+\end_inset
+
+ drevo 
+\begin_inset Formula $\Rightarrow\left|ET\right|=\left|VT\right|-1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Proof
+z indukcijo po številu vozlišč:
+\end_layout
+
+\begin_layout Proof
+Baza: 
+\begin_inset Formula $\left|VT=1\right|$
+\end_inset
+
+, 
+\begin_inset Formula $\left|EG\right|=0$
+\end_inset
+
+ (izolirano vozlišče je drevo)
+\end_layout
+
+\begin_layout Proof
+Korak: 
+\begin_inset Formula $T$
+\end_inset
+
+ poljubno drevo 
+\begin_inset Formula $\left|VG\right|\geq2$
+\end_inset
+
+.
+ 
+\begin_inset Formula $v\in VT$
+\end_inset
+
+ naj bo list v 
+\begin_inset Formula $T$
+\end_inset
+
+ po lemi 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "lem:Vsako-drevo-z"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+ Po I.
+ P.
+ je 
+\begin_inset Formula $\left|E\left(T-v\right)\right|=\left|V\left(T-v\right)\right|-1$
+\end_inset
+
+, po konstrukciji 
+\begin_inset Formula $T-v$
+\end_inset
+
+ pa je 
+\begin_inset Formula $\left|E\left(T-v\right)\right|=\left|ET\right|-1$
+\end_inset
+
+, 
+\begin_inset Formula $\left|V\left(T-v\right)\right|=\left|VT\right|-1$
+\end_inset
+
+, torej 
+\begin_inset Formula $\left|ET\right|=\left|VT\right|-1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Lemma
+\begin_inset CommandInset label
+LatexCommand label
+name "lem:lema3drevesa-enaciklu-g-e-povezan"
+
+\end_inset
+
+
+\begin_inset Formula $G$
+\end_inset
+
+ povezan, 
+\begin_inset Formula $e\in EG$
+\end_inset
+
+ leži na ciklu 
+\begin_inset Formula $\Rightarrow G-e$
+\end_inset
+
+ povezan.
+\end_layout
+
+\begin_layout Proof
+Trdimo, da v 
+\begin_inset Formula $G-e\exists u,v-$
+\end_inset
+
+pot.
+ Ker je 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan, 
+\begin_inset Formula $\exists u,v-$
+\end_inset
+
+pot 
+\begin_inset Formula $P$
+\end_inset
+
+ v 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Če 
+\begin_inset Formula $e\not\in P$
+\end_inset
+
+, ta pot obstaja tudi v 
+\begin_inset Formula $G-e$
+\end_inset
+
+.
+ Če 
+\begin_inset Formula $e\in P$
+\end_inset
+
+, očitno dobimo pot tako, da 
+\begin_inset Formula $e$
+\end_inset
+
+ v 
+\begin_inset Formula $P$
+\end_inset
+
+ nadomestimo s preostankom cikla, na katerem leži 
+\begin_inset Formula $e$
+\end_inset
+
+ v 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Lemma
+\begin_inset CommandInset label
+LatexCommand label
+name "lem:-povezan-."
+
+\end_inset
+
+
+\begin_inset Formula $G$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\Rightarrow\left|EG\right|\geq\left|VG\right|-1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ drevo, velja enakost po lemi 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "lem:-drevo"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, sicer 
+\begin_inset Formula $G$
+\end_inset
+
+ premore cikel.
+ Po lemi 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "lem:lema3drevesa-enaciklu-g-e-povezan"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ 
+\begin_inset Formula $\exists e\in EG\ni:G-e$
+\end_inset
+
+ povezan.
+ Odstranjevanje povezav iz ciklov ponavljamo, dokler ne dobimo drevesa 
+\begin_inset Formula $T$
+\end_inset
+
+.
+ Velja 
+\begin_inset Formula $VT=VG$
+\end_inset
+
+ (*) in 
+\begin_inset Formula $\left|ET\right|<\left|EG\right|$
+\end_inset
+
+ (**).
+ Torej:
+\begin_inset Formula $\left|VG\right|-1\overset{\text{(*)}}{=}\left|VT\right|-1\overset{\text{lema \ref{lem:-drevo}}}{=}\left|ET\right|\overset{\text{(**)}}{<}\left|EG\right|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Karakterizacija dreves.
+ NTSE za graf 
+\begin_inset Formula $G$
+\end_inset
+
+:
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:-drevo"
+
+\end_inset
+
+
+\begin_inset Formula $G$
+\end_inset
+
+ drevo
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:-pot-ZDB"
+
+\end_inset
+
+
+\begin_inset Formula $\forall u,v\in VG\exists!$
+\end_inset
+
+ 
+\begin_inset Formula $u,v-$
+\end_inset
+
+pot ZDB za vsak par vozlišč obstaja enolična pot
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:-povezan-"
+
+\end_inset
+
+
+\begin_inset Formula $G$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\wedge\forall e\in EG:e$
+\end_inset
+
+ most
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:-povezan"
+
+\end_inset
+
+
+\begin_inset Formula $G$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\wedge\left|EG\right|=\left|VG\right|-1$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Proof
+Dokazujemo ekvivalenco:
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\ref{enu:-drevo}\Rightarrow\ref{enu:-pot-ZDB}\right)$
+\end_inset
+
+ PDDRAA 
+\begin_inset Formula $\exists$
+\end_inset
+
+ dve različni 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti za neka 
+\begin_inset Formula $u,v\in VG$
+\end_inset
+
+.
+ Tedaj graf premore cikel 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ ni gozd 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ ni drevo 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+ PDDRAA 
+\begin_inset Formula $\nexists$
+\end_inset
+
+ 
+\begin_inset Formula $u,v-$
+\end_inset
+
+pot za neka 
+\begin_inset Formula $u,v\in VG\Rightarrow\Omega G\not=1\Rightarrow$
+\end_inset
+
+ ni drevo 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\ref{enu:-pot-ZDB}\Rightarrow\ref{enu:-povezan-}\right)$
+\end_inset
+
+ PDDRAA 
+\begin_inset Formula $\exists uv\in EG\ni:$
+\end_inset
+
+ ni most 
+\begin_inset Formula $\Rightarrow\exists u,v-$
+\end_inset
+
+pot v 
+\begin_inset Formula $G-e\Rightarrow\exists$
+\end_inset
+
+ dve različni 
+\begin_inset Formula $u,v-$
+\end_inset
+
+poti v 
+\begin_inset Formula $G$
+\end_inset
+
+ (
+\begin_inset Formula $uv$
+\end_inset
+
+ in tista v 
+\begin_inset Formula $G-e$
+\end_inset
+
+) 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\ref{enu:-povezan-}\Rightarrow\ref{enu:-povezan}\right)$
+\end_inset
+
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\Rightarrow G$
+\end_inset
+
+ povezan velja vedno.
+ Dokažimo 
+\begin_inset Formula $\left|EG\right|=\left|VG\right|-1$
+\end_inset
+
+ z indukcijo po številu vozlišč:
+\end_layout
+
+\begin_deeper
+\begin_layout Standard
+Baza: V 
+\begin_inset Formula $K_{2}$
+\end_inset
+
+ je edina povezava most in 
+\begin_inset Formula $\left|EK_{2}\right|=2-1=1=\left|VK_{2}\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Korak: 
+\begin_inset Formula $e\in EG$
+\end_inset
+
+ poljuben: 
+\begin_inset Formula $\Omega\left(G-e\right)=2$
+\end_inset
+
+, dve komponenti 
+\begin_inset Formula $G-e$
+\end_inset
+
+ naj bosta 
+\begin_inset Formula $G_{1}$
+\end_inset
+
+ in 
+\begin_inset Formula $G_{2}$
+\end_inset
+
+.
+ Slednja sta povezana in za njiju velja, da je vsaka povezava most in sta
+ manjša grafa.
+ Po I.
+ P.
+ velja 
+\begin_inset Formula $\left|EG_{1}\right|=\left|VG_{1}\right|-1$
+\end_inset
+
+ in 
+\begin_inset Formula $\left|EG_{2}\right|=\left|VG_{2}\right|-1$
+\end_inset
+
+.
+ Velja 
+\begin_inset Formula $\left|EG\right|=\left|EG_{1}\right|+\left|EG_{2}\right|+1=\left|VG_{1}\right|-1+\left|VG_{2}\right|-1+1=\left|VG_{1}\right|+\left|VG_{2}\right|-1=\left|VG\right|-1$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\ref{enu:-povezan}\Rightarrow\ref{enu:-pot-ZDB}\right)$
+\end_inset
+
+ PDDRAA 
+\begin_inset Formula $G$
+\end_inset
+
+ premore cikel 
+\begin_inset Formula $\overset{\text{lema \ref{lem:lema3drevesa-enaciklu-g-e-povezan}}}{\Rightarrow}$
+\end_inset
+
+ če mu odstranimo povezavo s cikla, bo ostal povezan.
+ Obenem zaradi velja 
+\begin_inset Formula $\left|E\left(G-e\right)\right|=\left|V\left(G-e\right)\right|-2$
+\end_inset
+
+, kar je v 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+ z lemo 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "lem:-povezan-."
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+ (
+\begin_inset Formula $G$
+\end_inset
+
+ povezan, toda ne velja implikacija).
+\end_layout
+
+\end_deeper
+\begin_layout Subsection
+Kaj je vpeto drevo grafa? Kateri grafi premorejo vpeta drevesa? Kako lahko
+ rekurzivno določimo število vpetih dreves povezanega grafa?
+\end_layout
+
+\begin_layout Definition*
+Podgraf 
+\begin_inset Formula $H$
+\end_inset
+
+ grafa 
+\begin_inset Formula $G$
+\end_inset
+
+ je vpet, če velja 
+\begin_inset Formula $VP=VG$
+\end_inset
+
+.
+ (Lahko pa ima ima 
+\begin_inset Formula $G$
+\end_inset
+
+ povezave, ki jih 
+\begin_inset Formula $H$
+\end_inset
+
+ nima).
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Vpeto drevo grafa 
+\begin_inset Formula $G$
+\end_inset
+
+ je vpet podgraf, ki je drevo.
+\end_layout
+
+\begin_layout Theorem*
+\begin_inset Formula $G$
+\end_inset
+
+ povezan 
+\begin_inset Formula $\Leftrightarrow G$
+\end_inset
+
+ premore vpeto drevo.
+\end_layout
+
+\begin_layout Proof
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ očitno.
+ 
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+: Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ drevo, je sam sebi vpeto drevo, sicer vsebuje cikel in iterativno uporabljamo
+ lemo 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "lem:lema3drevesa-enaciklu-g-e-povezan"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+, dokler ne konstruiramo vpetega drevesa.
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $\tau G$
+\end_inset
+
+ naj bo število vpetih dreves grafa 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Remark*
+\begin_inset Formula $T$
+\end_inset
+
+ drevo 
+\begin_inset Formula $\Rightarrow\tau T=1$
+\end_inset
+
+, 
+\begin_inset Formula $\Omega G>1\Rightarrow\tau G=0$
+\end_inset
+
+, 
+\begin_inset Formula $\tau C_{n}=n$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Z 
+\begin_inset Formula $G/e$
+\end_inset
+
+ označimo skrčitev povezave 
+\begin_inset Formula $e$
+\end_inset
+
+ v multigrafu
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+V multigrafu se ista povezava lahko pojavi večkrat.
+\end_layout
+
+\end_inset
+
+ 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ To je enako kot skrčitev povezave v grafu (
+\begin_inset Formula $G\backslash e$
+\end_inset
+
+), le da ne odstranimo večkratnih vzporednih povezav.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $G$
+\end_inset
+
+ povezan, 
+\begin_inset Formula $e\in EG\Rightarrow\tau G=\tau\left(G-e\right)+\tau\left(G/e\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Proof
+Naj bo 
+\begin_inset Formula $T$
+\end_inset
+
+ vpeto drevo v 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Naj bo 
+\begin_inset Formula $e\in EG$
+\end_inset
+
+ poljuben fiksen.
+ Vpetih dreves, za katere 
+\begin_inset Formula $e\not\in T$
+\end_inset
+
+, je 
+\begin_inset Formula $\tau\left(G-e\right)$
+\end_inset
+
+.
+ Vpetih dreves, za katere 
+\begin_inset Formula $e\in T$
+\end_inset
+
+, je 
+\begin_inset Formula $\tau\left(G/e\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Rekurzivno torej določamo število vpetih dreves grafa z zgornjo trditvijo
+ tako, da izbiramo poljubne povezave in jih ožamo ter odstranjujemo in nato
+ seštejemo 
+\begin_inset Formula $\tau$
+\end_inset
+
+ teh dveh operacij.
+\end_layout
+
+\begin_layout Subsection
+Kaj je Laplaceova matrika multigrafa? Kaj pravi Kirchoffov izrek o številu
+ vpetih dreves multigrafa?
+\end_layout
+
+\begin_layout Definition*
+Laplaceova matrika 
+\begin_inset Formula $LG$
+\end_inset
+
+ je kvadratna matrika dimenzije 
+\begin_inset Formula $n$
+\end_inset
+
+, katere vrstice in stolpci so indeksirani z vozlišči multigrafa 
+\begin_inset Formula $G$
+\end_inset
+
+ in velja
+\begin_inset Formula 
+\[
+LG_{ij}=\begin{cases}
+\deg_{G}v & ;i=j\\
+-\left(\text{število povezav med \ensuremath{v_{i}} in \ensuremath{v_{j}}}\right) & ;i\not=j
+\end{cases}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Theorem*
+Kirchoff: Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan multigraf, potem je 
+\begin_inset Formula $\tau G=\det\left(LG\text{ brez \ensuremath{i}tega stolpca in \ensuremath{i}te vrstice}\right)$
+\end_inset
+
+ za poljuben 
+\begin_inset Formula $i$
+\end_inset
+
+.
+ ZDB 
+\begin_inset Formula $\tau G=\det LG_{i,i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Kaj pomeni, da je graf Eulerjev? Kako karakteriziramo Eulerjeve grafe? Skicirajt
+e dokaz slednjega rezultata.
+\end_layout
+
+\begin_layout Definition*
+Sprehod v multigrafu je Eulerjev, če vsebuje vse povezave in sicer vsako
+ zgolj enkrat.
+ 
+\series bold
+Sklenjen
+\series default
+ Eulerjev sprehod je Eulerjev obhod.
+ Graf je Eulerjev, če premore Eulerjev obhod.
+\end_layout
+
+\begin_layout Theorem*
+\begin_inset Formula $G$
+\end_inset
+
+ Eulerjev 
+\begin_inset Formula $\Leftrightarrow\forall v\in VG:\deg_{G}v$
+\end_inset
+
+ sod.
+\end_layout
+
+\begin_layout Proof
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ očitno.
+ 
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+: 
+\begin_inset Formula $\forall v\in VG:\deg_{G}v$
+\end_inset
+
+ sod 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ premore cikel.
+ Indukcija po številu povezav:
+\end_layout
+
+\begin_layout Proof
+Baza: Izolirano vozlišče, multigraf 
+\begin_inset Formula $VG=\left\{ u,v\right\} ,EG=\left\{ uv,uv\right\} $
+\end_inset
+
+ in 
+\begin_inset Formula $C_{3}$
+\end_inset
+
+ so vsi Eulerjevi.
+\end_layout
+
+\begin_layout Proof
+Korak: Naj bo 
+\begin_inset Formula $\left|VG\right|\geq4$
+\end_inset
+
+.
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ gotovo premore cikel 
+\begin_inset Formula $C$
+\end_inset
+
+, ker je povezan in nima listov (ni drevo).
+ Naj bo 
+\begin_inset Formula $H\coloneqq G-EC$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\forall u\in VC:\deg_{H}u=\deg_{G}u-2$
+\end_inset
+
+, 
+\begin_inset Formula $\forall u\not\in VC:\deg_{H}u=\deg_{G}u$
+\end_inset
+
+ 
+\begin_inset Formula $\Rightarrow\forall v\in H:\deg_{H}v$
+\end_inset
+
+ sod 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ vsaka komponenta 
+\begin_inset Formula $H$
+\end_inset
+
+ je Eulerjev graf po I.
+ P., saj je manjši graf.
+ 
+\begin_inset Formula $G$
+\end_inset
+
+ je tudi Eulerjev, obhodimo ga lahko po ciklu 
+\begin_inset Formula $C$
+\end_inset
+
+; vsakič, ko naletimo na vozlišče, ki je del komponente 
+\begin_inset Formula $H$
+\end_inset
+
+, jo obhodimo in nadaljujemo pot po ciklu.
+\end_layout
+
+\begin_layout Paragraph*
+Fleuryjev algoritem
+\end_layout
+
+\begin_layout Standard
+sprejme graf in vrne obhod
+\end_layout
+
+\begin_layout Enumerate
+Začnemo v poljubni povezavi
+\end_layout
+
+\begin_layout Enumerate
+Ko povezavo prehodimo, jo izbrišemo
+\end_layout
+
+\begin_layout Enumerate
+Postopek nadaljujemo in pri tem pazimo le na to, da gremo na most le v primeru,
+ če ni druge možnosti.
+\end_layout
+
+\begin_layout Theorem*
+Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ Eulerjev graf, potem Fleuryjev algoritem vrne Eulerjev obhod.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+dodaj dekompozicijo in dekompozicijo v cikle
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Kdaj je graf Hamiltonov? Navedite in pojasnite potrebni pogoj z razpadom
+ grafa za obstoj Hamiltonovega cikla v grafu.
+\end_layout
+
+\begin_layout Definition*
+
+\series bold
+Hamiltonov cikel 
+\series default
+v grafu je cikel, ki vsebuje vsa vozlišča grafa ZDB Hamiltonov cikel je
+ vpet podgraf, ki je cikel.
+ 
+\series bold
+Graf je Hamiltonov
+\series default
+, če premore Hamiltonov cikel.
+ 
+\series bold
+Hamiltonova pot
+\series default
+ v grafu je pot, ki vsebuje vsa vozlišča grafa ZDB vpet podgraf, ki je pot.
+\end_layout
+
+\begin_layout Theorem*
+\begin_inset Formula $G$
+\end_inset
+
+ Hamiltonov, 
+\begin_inset Formula $S\subseteq VG\Rightarrow\Omega\left(G-S\right)\leq\left|S\right|$
+\end_inset
+
+.
+ (potreben pogoj za obstoj Hamiltonovega cikla)
+\end_layout
+
+\begin_layout Proof
+Naj bo 
+\begin_inset Formula $VG=\left\{ v_{1},\dots,v_{n}\right\} $
+\end_inset
+
+ Hamiltonov.
+ BSŠ naj bo 
+\begin_inset Formula $\left[v_{1},\dots,v_{n}\right]$
+\end_inset
+
+ Hamiltonov cikel.
+ Skica dokaza: Naj bosta 
+\begin_inset Formula $v_{i},v_{j}\in S,i<j$
+\end_inset
+
+.
+ Tedaj 
+\begin_inset Formula $G-S$
+\end_inset
+
+ razbije 
+\begin_inset Formula $G$
+\end_inset
+
+ na dva podgrafa: 
+\begin_inset Formula $K_{1}\left\{ v_{i+1},\dots,v_{j-1}\right\} $
+\end_inset
+
+ in 
+\begin_inset Formula $K_{2}=\left(VG\cap\left\{ v_{i},v_{j}\right\} \right)\cap K_{1}=\left\{ v_{j+1},\dots,v_{n},v_{1},\dots,v_{i-1}\right\} $
+\end_inset
+
+.
+ Če 
+\begin_inset Formula $i=j-1$
+\end_inset
+
+, je ena podgraf prazen, če 
+\begin_inset Formula $n=3$
+\end_inset
+
+ prav tako.
+ Podgrafa sta lahko povezana in tvorita skupno komponento, lahko pa nista
+ in tvorita dve komponenti.
+ Toda z odstranjevanjem 
+\begin_inset Formula $\left|S\right|$
+\end_inset
+
+ vozlišč iz cikla lahko napravimo največ 
+\begin_inset Formula $\left|S\right|$
+\end_inset
+
+ komponent.
+\end_layout
+
+\begin_layout Remark*
+Izrek uporabimo v kontrapoziciji: 
+\begin_inset Formula $\exists S\subseteq VG\ni:\Omega\left(G-S\right)>\left|S\right|\Rightarrow G$
+\end_inset
+
+ ni Hamiltonov.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $G$
+\end_inset
+
+ vsebuje prerezno vozlišče 
+\begin_inset Formula $\Rightarrow G$
+\end_inset
+
+ ni Hamiltonov.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $K_{m,n}$
+\end_inset
+
+ Hamiltonov 
+\begin_inset Formula $\Leftrightarrow m=n$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ Uporabimo izrek o potrebnem pogoju.
+ RAAPDD BSŠ 
+\begin_inset Formula $m>n$
+\end_inset
+
+: 
+\begin_inset Formula $\Omega\left(G-n\right)=m$
+\end_inset
+
+, kar vodi v 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ Očitno lahko skiciramo Hamiltonov cikel.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $G$
+\end_inset
+
+ dvodelen z razdelitvijo 
+\begin_inset Formula $\left(A,B\right)$
+\end_inset
+
+, 
+\begin_inset Formula $\left|A\right|\not=\left|B\right|\Rightarrow G$
+\end_inset
+
+ ni Hamiltonov.
+\end_layout
+
+\begin_layout Subsection
+Navedite Orejev zadostni pogoj za obstoj Hamiltonovega cikla v grafu.
+ Skicirajte dokaz tega izreka.
+\end_layout
+
+\begin_layout Theorem*
+Ore: 
+\begin_inset Formula $G$
+\end_inset
+
+ graf, 
+\begin_inset Formula $\left|VG\right|\geq3$
+\end_inset
+
+, 
+\begin_inset Formula $\left(\forall u,v\in VG:uv\not\in EG\Rightarrow\deg u+\deg v\geq\left|VG\right|\right)\Rightarrow G$
+\end_inset
+
+ Hamiltonov.
+ ZDB če za vsak par nesosednjih vozlišč v grafu z vsaj tremi vozlišči velja
+ 
+\begin_inset Formula $\deg u+\deg v\geq\left|VG\right|$
+\end_inset
+
+, je graf Hamiltonov.
+\end_layout
+
+\begin_layout Proof
+Dokaz z metodo najmanjšega protiprimera.
+ RAAPDD izrek ne velja.
+ Tedaj 
+\begin_inset Formula $\exists G$
+\end_inset
+
+, da predpostavka velja, zaključek pa ne.
+ Med vsemi takimi grafi izberimo tiste z najmanj vozlišči, izmed njih pa
+ enega izmed tistih, ki imajo največ povezav, in ga fiksiramo.
+ Naj bo to graf 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Zanj velja:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+\begin_inset Formula $\forall u,v\in VG:uv\not\in EG\Rightarrow\deg u+\deg v\geq\left|VG\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $G$
+\end_inset
+
+ ni Hamiltonov 
+\begin_inset Formula $\Rightarrow G$
+\end_inset
+
+ gotovo ni polni graf 
+\begin_inset Formula $\Rightarrow\exists u,v\in VG\ni:uv\not\in EG$
+\end_inset
+
+.
+ Naj bo 
+\begin_inset Formula $H$
+\end_inset
+
+ graf, da velja 
+\begin_inset Formula $VH\coloneqq VG$
+\end_inset
+
+ in 
+\begin_inset Formula $EH\coloneqq EG\cup uv$
+\end_inset
+
+.
+ Zanj še vedno velja prejšnja točka, zaradi izbire 
+\begin_inset Formula $G$
+\end_inset
+
+ (največ povezav) pa ni več protiprimer za izrek, zato je Hamiltonov.
+ Vsak Hamiltonov cikel v 
+\begin_inset Formula $H$
+\end_inset
+
+ vsebuje 
+\begin_inset Formula $uv$
+\end_inset
+
+, sicer bi obstajal že v 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Vseeno pa 
+\begin_inset Formula $G$
+\end_inset
+
+ premore Hamiltonovo pot, saj smo do cikla namreč dodali le eno povezavo.
+ Naj bo 
+\begin_inset Formula $\left[u=v_{1},v_{2},\dots,v_{n-1},v_{n}=v\right]$
+\end_inset
+
+ Hamiltonov cikel v 
+\begin_inset Formula $H$
+\end_inset
+
+.
+ Vpeljimo množici 
+\begin_inset Formula $S=\left\{ v_{i},uv_{i+1}\in EG\right\} $
+\end_inset
+
+ (ZDB predhodniki sosedov 
+\begin_inset Formula $u$
+\end_inset
+
+ na Hamiltonovi poti v 
+\begin_inset Formula $G$
+\end_inset
+
+) in 
+\begin_inset Formula $T=\left\{ v_{i},vv_{i}\in EG\right\} $
+\end_inset
+
+ (ZDB sosedje 
+\begin_inset Formula $v$
+\end_inset
+
+ na Hamiltonovi poti v 
+\begin_inset Formula $G$
+\end_inset
+
+).
+ Velja 
+\begin_inset Formula $\left|S\cup T\right|=\left|S\right|+\left|T\right|-\left|S\cap T\right|$
+\end_inset
+
+, torej 
+\begin_inset Formula $\left|S\cup T\right|+\left|S\cup T\right|=\left|S\right|+\left|T\right|=\deg_{G}u+\deg_{G}v\overset{\text{predpostavka}}{\geq}\left|VG\right|=n$
+\end_inset
+
+.
+ Toda ker je 
+\begin_inset Formula $\left|S\cup T\right|=\left|VG\right|$
+\end_inset
+
+, 
+\begin_inset Formula $\left|S\cap T\right|\not=\emptyset$
+\end_inset
+
+, torej ima 
+\begin_inset Formula $v$
+\end_inset
+
+ soseda iz 
+\begin_inset Formula $S$
+\end_inset
+
+ (recimo mu 
+\begin_inset Formula $v_{i}$
+\end_inset
+
+), torej lahko konstruiramo Hamiltonov cikel v 
+\begin_inset Formula $G$
+\end_inset
+
+: 
+\begin_inset Formula $\left[u=v_{1},v_{2},\dots,v_{i},v_{n}=v,v_{n-1},\dots,v_{i+1},v_{1}=u\right]$
+\end_inset
+
+ (
+\begin_inset Formula $v_{i+1}$
+\end_inset
+
+ je namreč po konstrukciji 
+\begin_inset Formula $S$
+\end_inset
+
+ sosed 
+\begin_inset Formula $u$
+\end_inset
+
+), kar vodi v 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:Kaj-so-ravninski"
+
+\end_inset
+
+Kaj so ravninski grafi? Kaj so lica ravninske vložitve grafa in čemu je
+ enaka vsota dolžin vseh lic ravninske vložitve grafa? Kako lahko omejimo
+ število povezav ravninskega grafa s pomočjo njegove ožine?
+\end_layout
+
+\begin_layout Definition*
+Graf 
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ lahko ga narišemo v ravnino tako, da se nobeni povezavi ne križata.
+ Ravninski graf skupaj z ustrezno risbo (vložitvijo) je graf, vložen v ravnino.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $K_{2,3}$
+\end_inset
+
+ je ravninski, 
+\begin_inset Formula $K_{3,3}$
+\end_inset
+
+ ni ravninski.
+\end_layout
+
+\begin_layout Theorem*
+Jordan: Sklenjena enostavna krivulja (t.
+ j.
+ taka, ki same sebe ne križa) v ravnini razdeli ravnino v notranjost, zunanjost
+ in krivuljo samo.
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski, vložen v ravnino.
+ Sklenjena območja v ravnini, dobljena tako, da iz risbe odstranimo točke,
+ ki ustrezajo vozliščem in povezavam, imenujemo 
+\series bold
+lica vložitve
+\series default
+.
+ S 
+\begin_inset Formula $FG$
+\end_inset
+
+ označimo množico lič vložitve.
+ Seveda je tudi zunanje/neomejeno območje lice.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\left|FQ_{3}\right|=6$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Remark*
+\begin_inset Formula $G$
+\end_inset
+
+ lahko vložimo v ravnino 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ lahko ga vložimo na sfero.
+\end_layout
+
+\begin_layout Definition*
+Dolžina lica 
+\begin_inset Formula $F$
+\end_inset
+
+ 
+\begin_inset Formula $(\text{\ensuremath{\ell F}}),$
+\end_inset
+
+je število povezav, ki jih prehodimo, ko obhodimo lice\SpecialChar endofsentence
+
+\end_layout
+
+\begin_layout Remark*
+Vsako drevo je ravninski graf.
+ Ima eno lice, katerega dolžina je 
+\begin_inset Formula $2\left|ET\right|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Ožina grafa 
+\begin_inset Formula $G$
+\end_inset
+
+, 
+\begin_inset Formula $gG$
+\end_inset
+
+, je dolžina najkrajšega cikla v 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ gozd, je 
+\begin_inset Formula $gG\coloneqq\infty$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ ravninsli graf, vložen v ravnino, velja
+\begin_inset Formula 
+\[
+\sum_{F\in FG}\ell F=2\left|EG\right|
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Claim*
+Naj 
+\begin_inset Formula $G$
+\end_inset
+
+ premore cikel.
+ Naj bo 
+\begin_inset Formula $F\in FG$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\ell F\geq gG$
+\end_inset
+
+.
+ 
+\begin_inset Formula $2\left|EG\right|=\sum_{F\in FG}\ell F\geq\sum_{F\in FG}gG=gG\left|FG\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Claim*
+Posledično: Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski graf z vsaj enim ciklom in je vložen v ravnino, je 
+\begin_inset Formula $\left|EG\right|\geq\frac{gG}{2}\left|FG\right|$
+\end_inset
+
+(*).
+\end_layout
+
+\begin_layout Subsection
+Kaj pravi Eulerjeva formula za ravninske grafe? Skicirajte njen dokaz.
+ Katere posledice Eulerjeve formule poznate?
+\end_layout
+
+\begin_layout Theorem*
+Eulerjeva formula: Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ ravninski vložen v ravnino, velja 
+\begin_inset Formula $\left|VG\right|-\left|EG\right|+\left|FG\right|=1+\Omega G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+Graf 
+\begin_inset Quotes gld
+\end_inset
+
+tri hiše
+\begin_inset Quotes grd
+\end_inset
+
+: 
+\begin_inset Formula $\left|VG\right|=15$
+\end_inset
+
+, 
+\begin_inset Formula $\left|EG\right|=18$
+\end_inset
+
+, 
+\begin_inset Formula $\left|FG\right|=7$
+\end_inset
+
+, 
+\begin_inset Formula $\Omega G=3$
+\end_inset
+
+, 
+\begin_inset Formula $15+16+7=1+3$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Proof
+Dokažimo najprej za povezan multigraf 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Dokazujemo 
+\begin_inset Formula $\left|VG\right|-\left|EG\right|+\left|FG\right|=2$
+\end_inset
+
+.
+ Indukcija po številu vozlišč:
+\end_layout
+
+\begin_layout Proof
+Baza: Izolirano vozlišče: 
+\begin_inset Formula $\left|VG\right|=1$
+\end_inset
+
+, 
+\begin_inset Formula $\left|EG\right|=0+z$
+\end_inset
+
+, 
+\begin_inset Formula $\left|FG\right|=1+z$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $z$
+\end_inset
+
+ število zank (za navaden graf 
+\begin_inset Formula $z=0$
+\end_inset
+
+).
+ Drži.
+\end_layout
+
+\begin_layout Proof
+Korak: Naj bo 
+\begin_inset Formula $e\in EG$
+\end_inset
+
+ poljubna.
+ Skrčimo jo (kot v multigrafu).
+ 
+\begin_inset Formula $\left|V\left(G/e\right)\right|=\left|VG\right|-1$
+\end_inset
+
+, 
+\begin_inset Formula $\left|E\left(G/e\right)\right|=\left|EG\right|-1$
+\end_inset
+
+, 
+\begin_inset Formula $\left|F\left(G/e\right)\right|=\left|FE\right|$
+\end_inset
+
+.
+ Velja 
+\begin_inset Formula $\left|VG\right|-\left|EG\right|+\left|FG\right|=2$
+\end_inset
+
+, saj po I.
+ P.
+ velja
+\begin_inset Formula $\left|V\left(G/e\right)\right|+1-\left|E\left(G/e\right)\right|-1+\left|F\left(G/e\right)\right|=2$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Sedaj dokažimo še za nepovezan multigraf 
+\begin_inset Formula $G$
+\end_inset
+
+ z 
+\begin_inset Formula $\Omega G$
+\end_inset
+
+ komponentami.
+ Grafu lahko dodamo 
+\begin_inset Formula $\Omega G-1$
+\end_inset
+
+ povezav, da ga povežemo, s čimer ne spremenimo niti 
+\begin_inset Formula $\left|FG\right|$
+\end_inset
+
+ niti 
+\begin_inset Formula $\left|VG\right|$
+\end_inset
+
+.
+ Če je 
+\begin_inset Formula $E$
+\end_inset
+
+ množica povezav, ki jo moramo dodati, velja 
+\begin_inset Formula $\left|VG\right|-\left|EG\cup E\right|+\left|FG\right|=2=\left|VG\right|-\left|EG\right|-\Omega G+1+\left|FG\right|\Rightarrow\left|VG\right|-\left|EG\right|+\left|FG\right|=2-1+\Omega G=1+\Omega G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Za ravninski graf z vsaj tremi vozlišči velja 
+\begin_inset Formula $\left|EG\right|\leq3\left|VG\right|-6$
+\end_inset
+
+, če je slednji brez trikotnikov, a ima cikel, pa celo 
+\begin_inset Formula $\left|EG\right|\leq2\left|VG\right|-4$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Dokažimo za povezan ravninski graf, sicer mu lahko samo dodamo povezave
+ in ga povežemo.
+ Ločimo primera:
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(G\text{ drevo}\right)$
+\end_inset
+
+ 
+\begin_inset Formula $\left|EG\right|=\left|VG\right|-1$
+\end_inset
+
+ (karakterizacija dreves) 
+\begin_inset Formula $\overset{?}{\leq}3\left|VG\right|-6$
+\end_inset
+
+.
+ Drži, kajti 
+\begin_inset Formula $\left|VG\right|\geq3$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(G\text{ premore cikel}\right)$
+\end_inset
+
+ Po Eulerjevi formuli velja 
+\begin_inset Formula 
+\[
+2=\left|VG\right|-\left|EG\right|+\left|FG\right|\overset{\text{(*) iz \ref{subsec:Kaj-so-ravninski}}}{\leq}\left|VG\right|-\left|EG\right|+\frac{2\left|EG\right|}{gG}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+2\leq\left|VG\right|-\left|EG\right|+\frac{2\left|EG\right|}{gG}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+2-\left|VG\right|\leq\left|EG\right|\left(\frac{2}{gG}-1\right)\quad\quad\quad\quad/^{-1}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left|VG\right|-2\geq\left|EG\right|\left(1-\frac{2}{gG}\right)=\left|EG\right|\left(\frac{gG-2}{gG}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left(\left|VG\right|-2\right)\frac{gG}{gG-2}\geq\left|EG\right|
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\frac{gG}{gG-2}$
+\end_inset
+
+ je največ 3 in to tedaj, ko graf premore trikotnik.
+ Če za vse večje 
+\begin_inset Formula $gG$
+\end_inset
+
+ bo ta ulomek manjši, torej lahko levo stran omejimo navzdol: 
+\begin_inset Formula $3\left|VG\right|-6\geq\left(\left|VG\right|-2\right)\frac{gG}{gG-2}\geq\left|EG\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Če pa graf ne premore trikotnika, a ima cikel, pa je 
+\begin_inset Formula $\frac{gG}{gG-2}=2$
+\end_inset
+
+ in levo stran strožje omejimo navzgor in velja 
+\begin_inset Formula $2\left|VG\right|-4\geq\left|EG\right|$
+\end_inset
+
+.
+ 
+\end_layout
+
+\end_deeper
+\begin_layout Subsection
+Kaj je kromatično število 
+\begin_inset Formula $\chi\left(G\right)$
+\end_inset
+
+ grafa 
+\begin_inset Formula $G$
+\end_inset
+
+? Pojasnite požrešni algoritem barvanja grafa.
+ Kako lahko z njegovo pomočjo navzgor omejimo 
+\begin_inset Formula $\chi\left(G\right)$
+\end_inset
+
+?
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $k-$
+\end_inset
+
+barvanje grafa 
+\begin_inset Formula $G$
+\end_inset
+
+ je preslikava 
+\begin_inset Formula $C:VG\to\left\{ 1..k\right\} $
+\end_inset
+
+, za katero velja 
+\begin_inset Formula $uv\in EG\Rightarrow Cu\not=Cv$
+\end_inset
+
+.
+ Kromatično število 
+\begin_inset Formula $\chi G$
+\end_inset
+
+ je najmanjši 
+\begin_inset Formula $k$
+\end_inset
+
+, za katerega najdemo 
+\begin_inset Formula $k-$
+\end_inset
+
+barvanje 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Za fiksen 
+\begin_inset Formula $i$
+\end_inset
+
+ je 
+\begin_inset Formula $\left\{ u\in VG;Cu=i\right\} $
+\end_inset
+
+ barvni razred, ki je neodvisna množica
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Za neodvisno množico 
+\begin_inset Formula $S\subseteq VG$
+\end_inset
+
+ velja 
+\begin_inset Formula $\forall u,v\in S:uv\not\in EG$
+\end_inset
+
+ ZDB je 
+\begin_inset Quotes gld
+\end_inset
+
+brez povezav
+\begin_inset Quotes grd
+\end_inset
+
+.
+ Glej vprašanje 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:Kaj-je-neodvisnostno"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+\end_layout
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\chi K_{n}=n$
+\end_inset
+
+, 
+\begin_inset Formula $\chi C_{n}=\begin{cases}
+2 & ;n\text{ sod}\\
+3 & ;n\text{ lih}
+\end{cases}$
+\end_inset
+
+, 
+\begin_inset Formula $\chi P_{5,2}=3$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Klično število grafa 
+\begin_inset Formula $G$
+\end_inset
+
+ označimo z 
+\begin_inset Formula $\omega G$
+\end_inset
+
+.
+ Velja 
+\begin_inset Formula $\omega G=\left|VH\right|$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $H$
+\end_inset
+
+ največji poln podgraf v 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Remark*
+\begin_inset Formula $\forall G\in\mathcal{G}:\chi G\geq\omega G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Požrešni algoritem barvanja: V poljubnem vrstnem redu zaporedno barvamo
+ vozlišča.
+ Posameznemu vozlišču priredimo najnižjo barvo, ki še ni uporabljena na
+ njegovih sosedih.
+\end_layout
+
+\begin_layout Fact*
+Vedno 
+\begin_inset Formula $\exists$
+\end_inset
+
+ vrstni red barvanja, da požrešni algoritem vrne barvanje s 
+\begin_inset Formula $\chi G$
+\end_inset
+
+ barvami.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $\forall G\in\mathcal{G}:\chi G\leq\Delta G+1$
+\end_inset
+
+ ZDB 
+\begin_inset Formula $\chi G$
+\end_inset
+
+ je kvečjemu 1 večji od največje stopnje v grafu.
+\end_layout
+
+\begin_layout Proof
+Naj bo 
+\begin_inset Formula $x_{1},\dots,x_{n}$
+\end_inset
+
+ poljuben vrstni red vozlišč.
+ Poženemo požrešni algoritem.
+ Na poljubnem 
+\begin_inset Formula $i$
+\end_inset
+
+tem koraku, ko barvamo 
+\begin_inset Formula $x_{i}$
+\end_inset
+
+, je kvečjemu 
+\begin_inset Formula $\deg_{G}x_{i}$
+\end_inset
+
+ barv, ki niso na razpolago, kar je 
+\begin_inset Formula $\leq\Delta G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $G$
+\end_inset
+
+ ni regularen 
+\begin_inset Formula $\Rightarrow\forall G\in\mathcal{G}:\chi G\leq\Delta G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Proof
+Naj bo 
+\begin_inset Formula $x$
+\end_inset
+
+ tisto vozlišče, ki največje stopnje (*).
+ Vzamemo ga kot koren za BFS in z BFS vozlišča zmečemo v zaporedje 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Nato požrešno barvamo v obratni smeri, kot jo določa 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Na vsakem koraku, razen na korenu, bomo imeli soseda, ki še ni pobarvan,
+ torej je kvečjemu 
+\begin_inset Formula $\deg_{G}x_{i}-1$
+\end_inset
+
+ barv, ki niso na razpolago, kar je 
+\begin_inset Formula $\leq\Delta G$
+\end_inset
+
+.
+ Na zadnjem koraku (koren) pa po predpostavki (*) na razpolago ni kvečjemu
+ 
+\begin_inset Formula $\Delta G-1$
+\end_inset
+
+ barv.
+\end_layout
+
+\begin_layout Theorem*
+Brooks: 
+\begin_inset Formula $G$
+\end_inset
+
+ povezan, 
+\begin_inset Formula $G$
+\end_inset
+
+ niti poln niti lihi cikel 
+\begin_inset Formula $\Rightarrow\chi G\leq\Delta G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Theorem*
+Naj bo 
+\begin_inset Formula $d_{1}\geq d_{2}\geq\cdots\geq d_{n}$
+\end_inset
+
+ zaporedje stopenj grafa 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Tedaj velja 
+\begin_inset Formula $\chi G\leq1+\max_{i=1}^{n}\left(\min\left\{ d_{i},i-1\right\} \right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Proof
+Poženimo požrešni algoritem barvanja v padajočem zaporedju stopenj.
+ Na 
+\begin_inset Formula $i$
+\end_inset
+
+tem barvamo vozlišče stopnje 
+\begin_inset Formula $d_{i}$
+\end_inset
+
+, zato imamo gotovo na voljo barvo iz 
+\begin_inset Formula $\left\{ 1..d_{i}+1\right\} $
+\end_inset
+
+.
+ Ker smo doslej pobarvali zgolj 
+\begin_inset Formula $i-1$
+\end_inset
+
+ vozlišč, imamo gotovo na voljo barvo iz 
+\begin_inset Formula $\left\{ 1..i\right\} $
+\end_inset
+
+.
+ Algoritem pobarva vozlišče z barvo 
+\begin_inset Formula $\leq\min\left\{ i,d_{i}+1\right\} $
+\end_inset
+
+.
+ Največja uporabljena barva 
+\begin_inset Formula $k$
+\end_inset
+
+ je 
+\begin_inset Formula $\leq\max_{i=1}^{n}\left\{ \min\left\{ d_{i}+1,i\right\} \right\} $
+\end_inset
+
+.
+ Torej 
+\begin_inset Formula $\chi G\leq1+\max_{i=1}^{n}\left(\min\left\{ d_{i},i-1\right\} \right)$
+\end_inset
+
+ (izven 
+\begin_inset Formula $\max$
+\end_inset
+
+ smo prišteli 1, znotraj 
+\begin_inset Formula $\min$
+\end_inset
+
+ smo od vseh elementov 1 odšteli).
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO XXX FIXME dokaz in trditev za barvanje ravninskega grafa s petimi barvami
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Kaj je kromatični indeks 
+\begin_inset Formula $\chi'\left(G\right)$
+\end_inset
+
+ grafa 
+\begin_inset Formula $G$
+\end_inset
+
+? Kaj pravi Vizingov izrek in kako na njegovi osnovi razdelimo vse grafe
+ v dva razreda?
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $k-$
+\end_inset
+
+barvanje povezav je preslikava 
+\begin_inset Formula $C:EG\to\left\{ 1..k\right\} \ni:uv,uw\in EG\Rightarrow C\left(uv\right)\not=C\left(uw\right)$
+\end_inset
+
+.
+ ZDB povezavi s skupnim krajiščem dobita različni barvi.
+ Kromatični indeks grafa 
+\begin_inset Formula $G$
+\end_inset
+
+ (oznaka 
+\begin_inset Formula $\chi'G$
+\end_inset
+
+) je najmanjši 
+\begin_inset Formula $k$
+\end_inset
+
+, za katerega 
+\begin_inset Formula $\exists k-$
+\end_inset
+
+barvanje grafa 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\chi'C_{n}=\begin{cases}
+2 & ;n\text{ sod}\\
+3 & ;n\text{ lih}
+\end{cases}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Theorem*
+Vizing: 
+\begin_inset Formula $\forall G\in\mathcal{G}:\Delta G\leq\chi'G\leq\Delta G+1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Prvi neenačaj je očiten, drugega ne bomo dokazali.
+\end_layout
+
+\begin_layout Definition*
+Graf 
+\begin_inset Formula $G$
+\end_inset
+
+ je razreda I, če je 
+\begin_inset Formula $\chi'G=\Delta G$
+\end_inset
+
+ oziroma razreda II, če je 
+\begin_inset Formula $\chi'G=\Delta G+1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $C_{2n}$
+\end_inset
+
+ so razreda 
+\begin_inset Formula $I$
+\end_inset
+
+, 
+\begin_inset Formula $C_{2n+1}$
+\end_inset
+
+ so razreda II, 
+\begin_inset Formula $K_{3}$
+\end_inset
+
+ je razreda II, 
+\begin_inset Formula $K_{4}$
+\end_inset
+
+ je razreda 
+\begin_inset Formula $I$
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO XXX FIXME dokaz, da so dvodelni grafi I, K_2k razreda I in K_2k+1 razreda
+ II.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:Kaj-je-neodvisnostno"
+
+\end_inset
+
+Kaj je neodvisnostno število grafa? Zanj podajte spodnjo mejo in zgornjo
+ mejo.
+ Opišite algoritem za izračun neodvisnostnega števila drevesa.
+\end_layout
+
+\begin_layout Definition*
+Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ graf in 
+\begin_inset Formula $I\subseteq VG$
+\end_inset
+
+, je 
+\begin_inset Formula $I$
+\end_inset
+
+ neodvisna 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in I:uv\not\in EG$
+\end_inset
+
+ ZDB če nobeni dve vozlišči v I nista sosednji v 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Moč največje neodvisne množice v 
+\begin_inset Formula $G$
+\end_inset
+
+ je neodvisnostno število 
+\begin_inset Formula $G$
+\end_inset
+
+, označeno z 
+\begin_inset Formula $\alpha G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\alpha K_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha C_{n}=\lfloor\frac{n}{2}\rfloor$
+\end_inset
+
+, 
+\begin_inset Formula $\alpha P_{5,2}=4$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $\forall G\in\mathcal{G}:\alpha G\cdot\chi G\geq\left|VG\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Proof
+Naj bo 
+\begin_inset Formula $\chi G=k$
+\end_inset
+
+ in 
+\begin_inset Formula $V_{1},\dots,V_{k}$
+\end_inset
+
+ barvni razredi nekega fiksnega barvanja s k barvami.
+ Slednji so neodvisne množice.
+ 
+\begin_inset Formula $\forall i\in\left\{ 1..k\right\} :\left|V_{i}\right|\leq\alpha G\Rightarrow\left|VG\right|=\sum_{i=1}^{k}\left|V_{i}\right|\leq\sum_{i=1}^{k}\alpha G=\alpha G\cdot k=\alpha G\chi G$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Corollary*
+Spodnja meja za neodvisnostno število 
+\begin_inset Formula $\alpha G\geq\frac{\left|VG\right|}{\chi G}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Claim*
+Zgornja meja za neodvisnostno število: 
+\begin_inset Formula $\alpha G\leq\left|VG\right|-\frac{\left|EG\right|}{\Delta G}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Naj bo 
+\begin_inset Formula $I$
+\end_inset
+
+ poljubna največja neodvisna množica v 
+\begin_inset Formula $G$
+\end_inset
+
+, torej
+\begin_inset Formula $\left|I\right|=\alpha G$
+\end_inset
+
+.
+ V 
+\begin_inset Formula $VG\setminus I$
+\end_inset
+
+ je vozlišč 
+\begin_inset Formula $\left|VG\right|-\alpha G$
+\end_inset
+
+.
+ Ker vozlišča v 
+\begin_inset Formula $I$
+\end_inset
+
+ medsebojno niso povezana, 
+\begin_inset Formula $\left|EG\right|\leq\left(\left|VG\right|-\alpha G\right)\cdot\Delta G\Rightarrow\left|EG\right|\leq\left|VG\right|\Delta G-\alpha G\Delta G\Rightarrow\alpha G\Delta G\leq\left|VG\right|\Delta G-\left|EG\right|\Rightarrow\alpha G\leq\left|VG\right|-\frac{\left|EG\right|}{\Delta G}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $Q_{d},d\geq1$
+\end_inset
+
+: Zgornja meja: 
+\begin_inset Formula $\alpha Q_{d}\leq\left|VQ_{d}\right|-\frac{\left|EQ_{d}\right|}{\Delta Q_{d}}=2^{d}-\frac{d2^{d-1}}{d}=2^{d}-2^{d-1}=2^{d-1}$
+\end_inset
+
+, Spodnja meja: 
+\begin_inset Formula $\alpha Q_{d}\geq\frac{\left|VQ_{d}\right|}{\chi Q_{d}}=\frac{2^{d}}{2}=2^{d-1}$
+\end_inset
+
+, torej 
+\begin_inset Formula $\alpha Q_{d}=2^{d-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph*
+Neodvisnostno število dreves
+\end_layout
+
+\begin_layout Standard
+Naj bo 
+\begin_inset Formula $T$
+\end_inset
+
+ drevo s poljubnim korenom 
+\begin_inset Formula $r\in VT$
+\end_inset
+
+.
+ Odslej na drevo glejmo 
+\begin_inset Formula $T$
+\end_inset
+
+ kot na drevo s korenom 
+\begin_inset Formula $r$
+\end_inset
+
+.
+ Za neodvisno množico 
+\begin_inset Formula $S$
+\end_inset
+
+ drevesa 
+\begin_inset Formula $T$
+\end_inset
+
+ in poljubno vozlišče 
+\begin_inset Formula $x\in VT$
+\end_inset
+
+ velja: 
+\begin_inset Formula $x\in S\Rightarrow$
+\end_inset
+
+ potomci (otroci) 
+\begin_inset Formula $x\not\in S$
+\end_inset
+
+.
+ Če pa 
+\begin_inset Formula $x\not\in S$
+\end_inset
+
+, pa 
+\begin_inset Formula $S$
+\end_inset
+
+ sme vsebovati potomce 
+\begin_inset Formula $x$
+\end_inset
+
+.
+ Z 
+\begin_inset Formula $Iv$
+\end_inset
+
+ označimo velikost največje neodvisne množice s korenom v 
+\begin_inset Formula $v\in VT$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Očitno velja 
+\begin_inset Formula $\alpha T=Ir$
+\end_inset
+
+.
+ Z rekurzivnim postopkom določimo 
+\begin_inset Formula $\text{\ensuremath{\alpha T}}$
+\end_inset
+
+ na tak način: 
+\begin_inset Formula 
+\[
+\alpha T=Ir=\text{\ensuremath{\max}\left\{  1+\sum_{v\in\text{vnuki/drugi potomci \ensuremath{r}}}Iv,\sum_{v\in\text{otroci/potomci }r}Iv\right\}  }
+\]
+
+\end_inset
+
+Formula je očitna.
+ Na vsakem rekurzivnem koraku lahko bodisi v množico izberemo 
+\begin_inset Formula $v$
+\end_inset
+
+ in ne izberemo njegovih potomcev, bodisi izberemo potomce, njega pa ne.
+ Z rekurzivnim algoritmom preverimo vse možnosti.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\alpha T_{n}$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $T_{n}$
+\end_inset
+
+ polno dvojiško drevo globine 
+\begin_inset Formula $n\in\mathbb{N}_{0}$
+\end_inset
+
+: Vpeljimo zaporedje 
+\begin_inset Formula $a_{n}=\alpha T_{n}$
+\end_inset
+
+ in velja: 
+\begin_inset Formula $\left(a_{n}\right)_{n\in\mathbb{N}_{0}}=\left[1,2,5,10,21,42,\dots\right]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+\begin_inset Formula $a_{0}=1$
+\end_inset
+
+, 
+\begin_inset Formula $a_{1}=2$
+\end_inset
+
+, za 
+\begin_inset Formula $n\geq2$
+\end_inset
+
+: 
+\begin_inset Formula $a_{n}=\begin{cases}
+2_{n-1}+1 & ;n\text{ sod}\\
+2_{n-1} & ;n\text{ lih}
+\end{cases}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Z indukcijo.
+ Baza sta 
+\begin_inset Formula $a_{0}$
+\end_inset
+
+ in 
+\begin_inset Formula $a_{1}$
+\end_inset
+
+.
+ Indukcijska predpostavka je dana v izreku.
+ ITD DOPIŠI DOKAZ.
+ 
+\begin_inset Quotes gld
+\end_inset
+
+DS2P FMF 2024-04-18
+\begin_inset Quotes grd
+\end_inset
+
+ stran 5.
+\end_layout
+
+\begin_layout Subsection
+\begin_inset CommandInset label
+LatexCommand label
+name "subsec:Kaj-je-dominacijsko"
+
+\end_inset
+
+Kaj je dominacijsko število grafa? Zanj podajte spodnjo mejo in zgornjo
+ mejo.
+ Kakšna je zveza med dominacijskim številom grafa in njegovega vpetega podgrafa?
+\end_layout
+
+\begin_layout Standard
+Naj bo 
+\begin_inset Formula $G$
+\end_inset
+
+ graf.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Neodvisna množica 
+\begin_inset Formula $G$
+\end_inset
+
+ je maksimalna, če ni prava podmnožica kakšne neodvisne množice v 
+\begin_inset Formula $G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+\begin_inset Formula $N_{G}\left(u\right)\coloneqq\left\{ v;uv\in EG\right\} $
+\end_inset
+
+ je soseščina vozlišča 
+\begin_inset Formula $u$
+\end_inset
+
+ (sosedje 
+\begin_inset Formula $u$
+\end_inset
+
+ v 
+\begin_inset Formula $G$
+\end_inset
+
+).
+ 
+\begin_inset Formula $N_{G}\left[u\right]\coloneqq N_{G}\left(u\right)\cup\left\{ u\right\} $
+\end_inset
+
+ je zaprta soseščina vozlišča 
+\begin_inset Formula $u$
+\end_inset
+
+.
+ 
+\begin_inset Formula $N_{G}\left[D\right]=\bigcup_{u\in D}N_{G}\left[u\right]$
+\end_inset
+
+ je zaprta soseščina množice vozlišč 
+\begin_inset Formula $D.$
+\end_inset
+
+ 
+\begin_inset Formula $D\subseteq VG$
+\end_inset
+
+ dominira 
+\begin_inset Formula $X\subseteq VG\Leftrightarrow X\subseteq N_{G}\left[D\right]$
+\end_inset
+
+.
+ Če 
+\begin_inset Formula $D$
+\end_inset
+
+ dominira 
+\begin_inset Formula $VG$
+\end_inset
+
+, pravimo, da je 
+\begin_inset Formula $D$
+\end_inset
+
+ dominantna množica grafa 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ ZDB 
+\begin_inset Formula $D$
+\end_inset
+
+ dominantna množica 
+\begin_inset Formula $G\Leftrightarrow\forall u\in VG:u\in D\vee\exists x\in D\ni:xu\in EG$
+\end_inset
+
+ ZDB vsako vozlišče je v 
+\begin_inset Formula $D$
+\end_inset
+
+ ali pa ima soseda iz 
+\begin_inset Formula $D$
+\end_inset
+
+.
+ Moč najmanjše dominacijske množice za 
+\begin_inset Formula $G$
+\end_inset
+
+ je dominacijsko število grafa 
+\begin_inset Formula $G$
+\end_inset
+
+, označeno z 
+\begin_inset Formula $\gamma G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Remark*
+Vsaka maksimalna neodvisna množica grafa je njegova dominantna množica.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\gamma K_{n}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\gamma C_{n}=\lceil\frac{n}{3}\rceil$
+\end_inset
+
+, 
+\begin_inset Formula $\gamma P_{5,2}=3$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Za vsak graf brez izoliranih vozlišč velja, da je 
+\begin_inset Formula $\lceil\frac{\left|VG\right|}{\Delta G+1}\rceil\leq\gamma G\leq\lfloor\frac{\left|VG\right|}{2}\rfloor$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Spodnja meja: Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ dominantna množica in 
+\begin_inset Formula $u\in D$
+\end_inset
+
+, potem 
+\begin_inset Formula $u$
+\end_inset
+
+ dominira 
+\begin_inset Formula $\leq\deg_{G}u+1$
+\end_inset
+
+ vozlišč, torej vsako vozlišče iz 
+\begin_inset Formula $D$
+\end_inset
+
+ dominira kvečjemu 
+\begin_inset Formula $\Delta G+1$
+\end_inset
+
+ vozlišč.
+ Ker 
+\begin_inset Formula $VG\subseteq N_{G}\left[D\right]$
+\end_inset
+
+ (unija zaprtih okolic vozlišč iz dominantne množice prekrije cel graf),
+ je 
+\begin_inset Formula $\left|D\right|\geq\frac{\left|VG\right|}{\Delta G+1}$
+\end_inset
+
+, kajti 
+\begin_inset Formula $\left|VG\right|=\left|N_{G}\left[D\right]\right|=\left|\bigcup_{u\in D}N_{G}\left[u\right]\right|\leq\sum_{u\in D}N\left(u\right)\leq\sum_{u\in D}$
+\end_inset
+
+
+\begin_inset Formula $\left(\Delta G+1\right)=\gamma G\left(\Delta G+1\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Zgornja meja: Naj bo 
+\begin_inset Formula $I$
+\end_inset
+
+ poljubna maksimalna neodvisna množica.
+ Vemo, da je 
+\begin_inset Formula $I$
+\end_inset
+
+ tedaj dominantna.
+ Če je 
+\begin_inset Formula $\left|I\right|\leq\frac{\left|VG\right|}{2}$
+\end_inset
+
+ smo dokazali, sicer vzemimo njen komplement 
+\begin_inset Formula $I'\coloneqq VG\setminus I$
+\end_inset
+
+.
+ Trdimo, da je 
+\begin_inset Formula $I'$
+\end_inset
+
+ dominantna.
+ Vzemimo poljubno 
+\begin_inset Formula $u\in G$
+\end_inset
+
+.
+ Če 
+\begin_inset Formula $u\in I'$
+\end_inset
+
+, dominira samega sebe, sicer, ker je 
+\begin_inset Formula $G$
+\end_inset
+
+ brez izoliranih vozlišč, ima 
+\begin_inset Formula $u$
+\end_inset
+
+ vsaj enega soseda, ker pa je 
+\begin_inset Formula $I$
+\end_inset
+
+ neodvisna, je ta sosed v 
+\begin_inset Formula $I'$
+\end_inset
+
+, torej je 
+\begin_inset Formula $I'$
+\end_inset
+
+ dominantna in ima 
+\begin_inset Formula $\leq\frac{\left|VG\right|}{2}$
+\end_inset
+
+ vozlišč in velja 
+\begin_inset Formula $\gamma G\leq\min\left\{ \left|I\right|,\left|I'\right|\right\} \leq\frac{\left|VG\right|}{2}$
+\end_inset
+
+, kajti 
+\begin_inset Formula $I\cup I'=VG$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+Enakost spodnje meje velja v 
+\begin_inset Formula $K_{n}$
+\end_inset
+
+, 
+\begin_inset Formula $C_{n}$
+\end_inset
+
+.
+ Enakost zgornje meje velja pri glavnikih 
+\begin_inset Formula $T_{n}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+dodaj 2-pakiranje in 2-pakirno število 
+\begin_inset Formula $\rho$
+\end_inset
+
+!
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Theorem*
+Če je 
+\begin_inset Formula $H$
+\end_inset
+
+ vpet podgraf 
+\begin_inset Formula $G$
+\end_inset
+
+, je 
+\begin_inset Formula $\gamma G\leq\gamma H$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Naj bo 
+\begin_inset Formula $D$
+\end_inset
+
+ minimalna dominantna množica za 
+\begin_inset Formula $H$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\left|D\right|=\gamma H$
+\end_inset
+
+.
+ Tedaj je 
+\begin_inset Formula $D$
+\end_inset
+
+ očitno tudi dominantna množica za 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Toda seveda se lahko zgodi, da je v 
+\begin_inset Formula $G$
+\end_inset
+
+ moč najti manjšo dominantno množico kot v 
+\begin_inset Formula $H$
+\end_inset
+
+, ker ima 
+\begin_inset Formula $G$
+\end_inset
+
+ lahko dodatne povezave.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO XXX FIXME vpeto drevo in 
+\begin_inset Formula $\gamma$
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Kaj je dominacijsko število grafa in kaj je celotno dominacijsko število
+ grafa? Kakšna je zveza med njima? Kakšna je povezava med dominacijskim
+ številom grafa in kromatičnim številom komplementa?
+\end_layout
+
+\begin_layout Standard
+Dominacijsko število grafa je definirano v vprašanju 
+\begin_inset CommandInset ref
+LatexCommand ref
+reference "subsec:Kaj-je-dominacijsko"
+plural "false"
+caps "false"
+noprefix "false"
+
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Dominantna množica 
+\begin_inset Formula $D$
+\end_inset
+
+ grafa 
+\begin_inset Formula $G$
+\end_inset
+
+ je povezana, če inducira povezan podgraf, neodvisna, če inducira podgraf
+ brez povezav in 
+\series bold
+celotna
+\series default
+, če ima vsako vozlišče iz 
+\begin_inset Formula $VG$
+\end_inset
+
+ soseda v 
+\begin_inset Formula $D$
+\end_inset
+
+ (tudi vozlišča iz 
+\begin_inset Formula $D$
+\end_inset
+
+).
+ Velikost najmanjše povezane dominantne množice označimo z 
+\begin_inset Formula $\gamma_{c}G$
+\end_inset
+
+, neodvisne z 
+\begin_inset Formula $\gamma_{i}G$
+\end_inset
+
+ in 
+\series bold
+celotne
+\series default
+ z 
+\begin_inset Formula $\gamma_{t}G$
+\end_inset
+
+ (connected, independent in total).
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\gamma_{t}K_{n}=2$
+\end_inset
+
+, 
+\begin_inset Formula $\gamma_{t}P_{n}=\lfloor\frac{n}{2}\rfloor+\lceil\frac{n}{2}\rceil-\lfloor\frac{n}{4}\rfloor$
+\end_inset
+
+ (gledamo parčke, ki se dominirajo).
+\end_layout
+
+\begin_layout Theorem*
+Za vsak graf brez izoliranih vozlišč velja 
+\begin_inset Formula $\gamma G\leq\gamma_{t}G\leq2\gamma G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Spodnja meja je očitna, kajti biti celotna dominantna množica je strožji
+ pogoj kot biti dominantna množica.
+ Dokažimo 
+\begin_inset Formula $\gamma_{t}G\leq2\gamma G$
+\end_inset
+
+: Naj bo 
+\begin_inset Formula $D$
+\end_inset
+
+ poljubna dominantna množica 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Vsakemu 
+\begin_inset Formula $x\in D$
+\end_inset
+
+ priredimo nekega soseda od 
+\begin_inset Formula $x$
+\end_inset
+
+, recimo 
+\begin_inset Formula $x'$
+\end_inset
+
+ in ga dodajmo v 
+\begin_inset Formula $\hat{D}$
+\end_inset
+
+, torej 
+\begin_inset Formula $\hat{D}=D\cup\left\{ x';x\in D\right\} $
+\end_inset
+
+, s čimer dominantno množico spremenimo v celotno tako, da ji kvečjemu podvojimo
+ število vozlišč.
+\end_layout
+
+\begin_layout Example*
+Enakost spodnje meje se pojavi pri 
+\begin_inset Formula $\gamma C_{4}=\gamma_{t}C_{4}=2$
+\end_inset
+
+, neenakost pa pri recimo 
+\begin_inset Formula $\gamma G_{3}=2\not=4=\gamma_{t}Q_{3}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Kaj je grupoid, polgrupa, monoid, grupa? Kako v monoidu izračunamo inverz
+ produkta obrnljivih elementov? Kako definiramo potence v monoidu in kateri
+ računski zakoni veljajo zanje?
+\end_layout
+
+\begin_layout Definition*
+Če uvedemo binarno operacijo 
+\begin_inset Formula $f$
+\end_inset
+
+ na množici 
+\begin_inset Formula $A$
+\end_inset
+
+ takole: 
+\begin_inset Formula $f:A\times A\to A,f$
+\end_inset
+
+ preslikava, je par 
+\begin_inset Formula $\left(A,f\right)$
+\end_inset
+
+ 
+\series bold
+grupoid
+\series default
+.
+ ZDB zahtevamo, da je operacija preslikava (domena je enaka definicijskemu
+ območju) in s tem zaprtost operacije.
+ Dvojiški operator 
+\begin_inset Formula $f\left(a,b\right)$
+\end_inset
+
+ pišimo kot 
+\begin_inset Formula $a\cdot b$
+\end_inset
+
+.
+ 
+\end_layout
+
+\begin_layout Definition*
+Če 
+\begin_inset Formula $\forall a,b,c\in A:\left(a\cdot b\right)\cdot c=a\cdot\left(b\cdot c\right)$
+\end_inset
+
+ (asociativnost), pravimo, da je 
+\begin_inset Formula $\left(A,\cdot\right)$
+\end_inset
+
+ 
+\series bold
+podgrupa
+\series default
+ ZDB asocietiven grupoid.
+\end_layout
+
+\begin_layout Definition*
+Enota je element 
+\begin_inset Formula $e\in A\ni:\forall a\in A:a\cdot e=e\cdot a=a$
+\end_inset
+
+.
+ Polgrupi z enoto pravimo 
+\series bold
+monoid
+\series default
+.
+ 
+\end_layout
+
+\begin_layout Definition*
+Monoidu, v katerem so vsi elementi obrnljivi, pravimo 
+\series bold
+grupa
+\series default
+ ZDB 
+\series bold
+
+\begin_inset Formula $\forall a\in A\exists b\in A\ni:a\cdot b=b\cdot a=e$
+\end_inset
+
+
+\series default
+.
+\end_layout
+
+\begin_layout Remark*
+Ker so inverzi v monoidu enolični (dokaz pri LA), inverz 
+\begin_inset Formula $a$
+\end_inset
+
+ označimo z 
+\begin_inset Formula $a^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\left(\mathcal{G},\square\right)$
+\end_inset
+
+ monoid, 
+\begin_inset Formula $\left(\mathbb{R}_{0}^{+}=\left\{ x\in\mathbb{R};x\geq0\right\} ,\max\right)$
+\end_inset
+
+ monoid, množica vseh nizov/seznamov s concat operacijo je monoid.
+\end_layout
+
+\begin_layout Theorem*
+Če sta 
+\begin_inset Formula $a$
+\end_inset
+
+ in 
+\begin_inset Formula $b$
+\end_inset
+
+ v monoidu obrnljiva, je obrnljiv tudi njun produkt in velja 
+\begin_inset Formula $\left(a\cdot b\right)^{-1}=b^{-1}\cdot a^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+\begin_inset Formula $\left(a\cdot b\right)\cdot\left(b^{-1}\cdot a^{-1}\right)=a\cdot\left(b\cdot b^{-1}\right)\cdot a^{-1}=a\cdot e\cdot a^{-1}=a\cdot a^{-1}=e$
+\end_inset
+
+ in podobno 
+\begin_inset Formula $\left(b^{-1}a^{-1}\right)\left(ab\right)=e$
+\end_inset
+
+, torej 
+\begin_inset Formula $\left(b^{-1}a^{-1}\right)\left(ab\right)=\left(ab\right)\left(b^{-1}a^{-1}\right)=e\Rightarrow ab=\left(b^{-1}a^{-1}\right)$
+\end_inset
+
+ po definiciji inverza.
+\end_layout
+
+\begin_layout Corollary*
+Če so 
+\begin_inset Formula $a_{1},\dots,a_{k}$
+\end_inset
+
+ obrnljivi elementi monoida, je 
+\begin_inset Formula $\left(a_{1}\cdots a_{k}\right)$
+\end_inset
+
+ obrnljiv element monoida in velja 
+\begin_inset Formula $\left(a_{1}\cdots a_{k}\right)^{-1}=a_{k}^{-1}\cdots a_{1}^{-1}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Dokaz z indukcijo: Baza: 
+\begin_inset Formula $k=2$
+\end_inset
+
+ velja, Korak: predpostavimo 
+\begin_inset Formula $\left(a_{1}\cdots a_{k}\right)^{-1}=a_{k}^{-1}\cdots a_{1}^{-1}$
+\end_inset
+
+.
+ Množimo z desne z 
+\begin_inset Formula $a_{k+1}$
+\end_inset
+
+ in smo dokazali.
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $\left(A,\cdot\right)$
+\end_inset
+
+ monoid, 
+\begin_inset Formula $n\in\mathbb{N}_{0}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $a^{0}\coloneqq e$
+\end_inset
+
+, 
+\begin_inset Formula $a^{n}=a\cdot a^{n-1}$
+\end_inset
+
+ za 
+\begin_inset Formula $n\geq1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Corollary*
+Velja torej 
+\begin_inset Formula $a^{n}a^{m}=a^{n+m}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(a^{n}\right)^{m}=a^{nm}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(a^{-1}\right)^{n}=a^{-1}\cdots n-\text{krat}\cdots a^{-1}=\left(a\cdots n-\text{krat}\cdots a\right)^{-1}=\left(a^{n}\right)^{-1}$
+\end_inset
+
+.
+ Torej za obrnljiv 
+\begin_inset Formula $a$
+\end_inset
+
+ velja 
+\begin_inset Formula $\left(a^{n}\right)^{-1}=\left(a^{-1}\right)^{n}$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+dodaj podgrupe etc
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Subsection
+Kaj je red elementa v grupi? Kaj je pravilo krajšanja v grupi? Dokažite
+ ga.
+ Kako se pravilo krajšanja odraža v Cayleyevi tabeli grupe?
+\end_layout
+
+\begin_layout Definition*
+Cayleyeva tabela za grupoid 
+\begin_inset Formula $\left(A,\cdot\right)$
+\end_inset
+
+ je kvadratna tabela širine 
+\begin_inset Formula $\left|A\right|$
+\end_inset
+
+, kjer ima 
+\begin_inset Formula $i,j-$
+\end_inset
+
+ta celica vrednost 
+\begin_inset Formula $a_{i}\cdot a_{j}$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $a_{k}$
+\end_inset
+
+ 
+\begin_inset Formula $k-$
+\end_inset
+
+ti element množice 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Pač izberemo si neko linearno urejenost 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $\left(G,\cdot\right)$
+\end_inset
+
+ končna grupa in 
+\begin_inset Formula $a\in G$
+\end_inset
+
+.
+ Tedaj je red elementa 
+\begin_inset Formula $a$
+\end_inset
+
+ najmanjši 
+\begin_inset Formula $n\in\mathbb{N}\ni:a^{n}=e$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Dirichletovo načelo: 
+\begin_inset Formula $\exists n,m\in\left[k+1\right],n\not=m:a^{n}=a^{m}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Proof
+BSŠ 
+\begin_inset Formula $n<m$
+\end_inset
+
+.
+ 
+\begin_inset Formula $a^{n}=a^{m}\Rightarrow a^{n}\left(a^{m}\right)^{-1}=a^{m}\left(a^{m}\right)^{-1}\Rightarrow a^{n-m}=e$
+\end_inset
+
+ 
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Theorem*
+Pravilo krajšanja: Če je 
+\begin_inset Formula $G$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+Ko omenimo algebrajsko strukturo, občasno omenimo le množico, ko je iz konteksta
+ operacija implicitno znana.
+\end_layout
+
+\end_inset
+
+ grupa, 
+\begin_inset Formula $a,b,c\in G$
+\end_inset
+
+, velja 
+\begin_inset Formula $ab=ac\Rightarrow b=c$
+\end_inset
+
+ in 
+\begin_inset Formula $ba=ca\Rightarrow b=c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Množimo obe strani z leve ali desne z inverzom 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Corollary*
+V Cayleyevi tabeli grupe se v vsaki vrstici in v vsakem stolpcu pojavijo
+ vsi elementi grupe.
+\end_layout
+
+\begin_layout Subsection
+Kaj je permutacijska grupa? Kaj je simetrična grupa 
+\begin_inset Formula $S_{n}$
+\end_inset
+
+ in kaj je alternirajoča grupa 
+\begin_inset Formula $A_{n}$
+\end_inset
+
+? Kaj pravi Cayleyev izrek (o univerzalnosti permutacijskih grup) in kakšna
+ je ideja njegovega dokaza?
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $A$
+\end_inset
+
+ množica.
+ Permutacija 
+\begin_inset Formula $A$
+\end_inset
+
+ je bijekcija 
+\begin_inset Formula $A\to A$
+\end_inset
+
+.
+ Permutacijska grupa je množica nekaj permutacij 
+\begin_inset Formula $A$
+\end_inset
+
+, ki ga komponiranje tvorijo grupo.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Simetrična grupa: 
+\begin_inset Formula $S_{n}\coloneqq\left\{ \pi;\pi:\left\{ 1..n\right\} \to\left\{ 1..n\right\} \text{ bijekcija}\right\} $
+\end_inset
+
+.
+ Alternirajoča grupa: 
+\begin_inset Formula $A_{n}\coloneqq\left\{ \pi\in S_{n};\pi\text{ soda}\right\} $
+\end_inset
+
+.
+ Permutacija je soda, če je v zapisu z disjunktnimi cikli vsota dolžin ciklov,
+ ki ji odštejemo število ciklov, soda.
+\end_layout
+
+\begin_layout Theorem*
+\begin_inset Formula $\left|A_{n}\right|=\frac{n!}{2}$
+\end_inset
+
+.
+ 
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+DOKAŽI TODO XXX FIXME
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Naj bosta 
+\begin_inset Formula $\left(G,\circ\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $\left(H,*\right)$
+\end_inset
+
+ grupi.
+ 
+\begin_inset Formula $f:G\to H$
+\end_inset
+
+ je hm
+\begin_inset Formula $\varphi\Leftrightarrow\forall x,y\in G:f\left(x\circ y\right)=fx*fy$
+\end_inset
+
+.
+ Če je 
+\begin_inset Formula $f$
+\end_inset
+
+ tudi bijekcija, je 
+\begin_inset Formula $f:G\to H$
+\end_inset
+
+ izomorfizem.
+ Grupi sta izomorfni, če obstaja izomorfizem med njima.
+ Tedaj pišemo 
+\begin_inset Formula $G\approx H$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Theorem*
+Cayley: Vsaka grupa je izomorfna neki permutacijski grupi.
+\end_layout
+
+\begin_layout Proof
+(skica dokaza) Naj bo 
+\begin_inset Formula $\left(G,\cdot\right)$
+\end_inset
+
+ grupa.
+ Vsakemu elementu grupe priredimo preslikavo 
+\begin_inset Formula $g\in G\mapsto\alpha_{g}:G\to G$
+\end_inset
+
+, ki slika takole: 
+\begin_inset Formula $\forall x\in G:\alpha_{k}x=g\cdot x$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\alpha_{g}$
+\end_inset
+
+ je permutacija 
+\begin_inset Formula $G$
+\end_inset
+
+, saj je bijektivna, ker velja pravilo krajšanja: 
+\begin_inset Formula $x\not=y\Rightarrow\alpha_{g}x\overset{\text{pravilo krajšanja}}{=}gx\not=gy=\alpha_{g}y$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\left\{ \alpha_{g};\forall g\in G\right\} $
+\end_inset
+
+ tvori permutacijsko grupo, ki je izomorfna izvorni grupi: 
+\begin_inset Formula $\left(\left\{ \alpha_{g};\forall g\in G\right\} ,\circ\right)\approx\left(G,\cdot\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Kaj so odseki grupe 
+\begin_inset Formula $G$
+\end_inset
+
+ po podgrupi 
+\begin_inset Formula $H$
+\end_inset
+
+? Kdaj je 
+\begin_inset Formula $aH=H$
+\end_inset
+
+ in kdaj je 
+\begin_inset Formula $aH=Ha$
+\end_inset
+
+? Kaj je vsebina Lagrangeovega izreka o moči podgrup in končnih grup?
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $\left(G,\cdot\right)$
+\end_inset
+
+ grupa in 
+\begin_inset Formula $H$
+\end_inset
+
+ podgrupa
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+prosim definiraj podgrupo
+\end_layout
+
+\end_inset
+
+ 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Če je 
+\begin_inset Formula $a\in G$
+\end_inset
+
+, definiramo 
+\begin_inset Formula $aH=\left\{ ah,h\in H\right\} $
+\end_inset
+
+ (desni odsek podgrupe 
+\begin_inset Formula $H$
+\end_inset
+
+) in 
+\begin_inset Formula $Ha=\left\{ ha,h\in H\right\} $
+\end_inset
+
+ (levi odsek podgrupe 
+\begin_inset Formula $H$
+\end_inset
+
+).
+\end_layout
+
+\begin_layout Theorem*
+Naj bo 
+\begin_inset Formula $G$
+\end_inset
+
+ grupa in 
+\begin_inset Formula $a,b\in G$
+\end_inset
+
+ in 
+\begin_inset Formula $H$
+\end_inset
+
+ podgrupa 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Tedaj veljajo naslednje lastnosti:
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:"
+
+\end_inset
+
+
+\begin_inset Formula $a\in aH$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:-1"
+
+\end_inset
+
+
+\begin_inset Formula $aH=H\Leftrightarrow a\in H$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $aH=bH\nabla aH\cap bH=\emptyset$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $aH=bH\Leftrightarrow a^{-1}b\in H$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left|aH\right|=\left|bH\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset CommandInset label
+LatexCommand label
+name "enu:-2"
+
+\end_inset
+
+
+\begin_inset Formula $aH=Ha\Leftrightarrow H=aHa^{-1}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $aH$
+\end_inset
+
+ podgrupa 
+\begin_inset Formula $G\Leftrightarrow a\in H$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Proof
+Dokažimo trditve:
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+\begin_inset Formula $e\in H\Rightarrow a\cdot e=a\in aH$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ 
+\begin_inset Formula $a\overset{\ref{enu:}}{\in}aH=H\Rightarrow a\in H$
+\end_inset
+
+, 
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ Najprej ena vsebovanost: 
+\begin_inset Formula $\forall x\in aH\exists h\in H\ni:x=ah$
+\end_inset
+
+, ker 
+\begin_inset Formula $a\in H$
+\end_inset
+
+, je po zaprtosti podgrupe 
+\begin_inset Formula $H$
+\end_inset
+
+ 
+\begin_inset Formula $ah=x\in H\Rightarrow aH\subseteq H$
+\end_inset
+
+.
+ Nato še druga vsebovanost: 
+\begin_inset Formula $\forall x\in H:a^{-1}x\in H\Rightarrow a\left(a^{-1}x\right)=x\in aH\Rightarrow H\subseteq aH$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Imejmo 
+\begin_inset Formula $aH$
+\end_inset
+
+, 
+\begin_inset Formula $bH$
+\end_inset
+
+.
+ Če je 
+\begin_inset Formula $aH=bH$
+\end_inset
+
+, smo dokazali, sicer 
+\begin_inset Formula $\exists x\in aH\cup bH$
+\end_inset
+
+.
+ Ker 
+\begin_inset Formula $x\in aH\Rightarrow\exists h'\in H\ni:x=ah'$
+\end_inset
+
+.
+ Ker 
+\begin_inset Formula $x\in bH\exists h''\in H\ni:x=bh''$
+\end_inset
+
+.
+ Toree velja 
+\begin_inset Formula $x=ah'=bh''$
+\end_inset
+
+.
+ Množimo s 
+\begin_inset Formula $h'^{-1}\Rightarrow a=bh''h'^{-1}\Rightarrow aH=bh''\left(h'^{-1}H\right)\overset{\ref{enu:-1}}{=}bh''H\overset{\ref{enu:-1}}{=}bH$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+...
+\end_layout
+
+\begin_layout Enumerate
+Očitno (pravilo krajšanja)
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ 
+\begin_inset Formula $aH=Ha\Rightarrow aHa^{-1}=Haa^{-1}=H$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ 
+\begin_inset Formula $H=aHa^{-1}\overset{\cdot a^{-1}}{\Rightarrow}a^{-1}H=Ha$
+\end_inset
+
+.
+ Ker je 
+\begin_inset Formula $a\in aH$
+\end_inset
+
+, je v 
+\begin_inset Formula $a^{-1}\in aH$
+\end_inset
+
+ (grupa), ker je 
+\begin_inset Formula $a^{-1}\in aH\cap a^{-1}H$
+\end_inset
+
+, je 
+\begin_inset Formula $aH=a^{-1}H$
+\end_inset
+
+, torej 
+\begin_inset Formula $a^{-1}H=Ha\Rightarrow aH=Ha$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Theorem*
+Lagrange: Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ končna grupa in 
+\begin_inset Formula $H$
+\end_inset
+
+ podgrupa 
+\begin_inset Formula $G$
+\end_inset
+
+, potem 
+\begin_inset Formula $\left|H\right|$
+\end_inset
+
+ deli 
+\begin_inset Formula $\left|G\right|$
+\end_inset
+
+.
+ Nadalje je število levih/desnih odsekov po 
+\begin_inset Formula $H$
+\end_inset
+
+ enako 
+\begin_inset Formula $\frac{\left|G\right|}{\left|H\right|}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+Naj bodo 
+\begin_inset Formula $a_{1}H,a_{2}H,\dots,a_{n}H$
+\end_inset
+
+ različni odseki.
+ Tedaj 
+\begin_inset Formula $\left|G\right|\overset{\ref{enu:}}{=}\left|a_{1}H\cup\cdots\cup a_{n}H\right|\overset{\text{disjunktni odseki}}{=}\sum_{i=1}^{k}\left|a_{i}H\right|=k\left|H\right|\Rightarrow k=\frac{\left|G\right|}{\left|H\right|}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Corollary*
+Če je 
+\begin_inset Formula $G$
+\end_inset
+
+ končna grupa in 
+\begin_inset Formula $a\in G$
+\end_inset
+
+, potem 
+\begin_inset Formula $\red a$
+\end_inset
+
+ deli 
+\begin_inset Formula $\left|G\right|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Proof
+\begin_inset Formula $\left\langle a\right\rangle =\left\{ a^{k};k\in\mathbb{Z}\right\} =\left\{ e,a,\dots,a^{n-1}\right\} \overset{\text{Lagrange}}{\Rightarrow}\left\langle a\right\rangle $
+\end_inset
+
+ je gotovo podgrupa 
+\begin_inset Formula $G\Rightarrow n$
+\end_inset
+
+ deli 
+\begin_inset Formula $\left|G\right|$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Kaj je podgrupa edinka? Navedite nekaj primerov podgrup edink.
+ Kaj je faktorska grupa 
+\begin_inset Formula $G/H$
+\end_inset
+
+ in kaj pomeni, da je operacija v 
+\begin_inset Formula $G/H$
+\end_inset
+
+ dobro definirana?
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $H$
+\end_inset
+
+ podgrupa grupe 
+\begin_inset Formula $G$
+\end_inset
+
+.
+ Tedaj rečemo, da je 
+\begin_inset Formula $H$
+\end_inset
+
+ podgrupa edinka, če velja 
+\begin_inset Formula $\forall a\in G:aH=Ha\overset{\ref{enu:-2}}{\Leftrightarrow}\forall a\in G:aHa^{-1}=H$
+\end_inset
+
+.
+ Oznaka: 
+\begin_inset Formula $H\triangleleft G$
+\end_inset
+
+.
+ Če sta 
+\begin_inset Formula $G$
+\end_inset
+
+ in 
+\begin_inset Formula $\left\{ e\right\} $
+\end_inset
+
+, kjer je 
+\begin_inset Formula $e$
+\end_inset
+
+ enota v 
+\begin_inset Formula $G$
+\end_inset
+
+, edini edinki 
+\begin_inset Formula $G$
+\end_inset
+
+, je 
+\begin_inset Formula $G$
+\end_inset
+
+ enostavna.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Center grupe 
+\begin_inset Formula $G\sim ZG\coloneqq\left\{ a\in G;\forall x\in G:ax=xa\right\} $
+\end_inset
+
+ ZDB taki elementi 
+\begin_inset Formula $G$
+\end_inset
+
+, ki komutirajo z vsemi.
+\end_layout
+
+\begin_layout Example*
+V Abelovi grupi je vsaka podgrupa edinka.
+ 
+\begin_inset Formula $SL_{2}\mathbb{R}\triangleleft GL_{2}\mathbb{R}$
+\end_inset
+
+
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+NAPIŠI KAJ JE TO SPECIAL LINEAR ITD
+\end_layout
+
+\end_inset
+
+.
+ 
+\begin_inset Formula $ZG\triangleleft G$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $H\triangleleft G$
+\end_inset
+
+.
+ 
+\begin_inset Formula $G/H\coloneqq\left\{ aH;a\in G\right\} $
+\end_inset
+
+.
+ V množico 
+\begin_inset Formula $G/H$
+\end_inset
+
+ vpeljemo operacijo 
+\begin_inset Formula $\left(aH\right)\left(bH\right)=\left(ab\right)H$
+\end_inset
+
+ (*).
+\end_layout
+
+\begin_layout Theorem*
+Faktorske grupe: Če je 
+\begin_inset Formula $H\triangleleft G$
+\end_inset
+
+, je 
+\begin_inset Formula $G/H$
+\end_inset
+
+ grupa za (*).
+\end_layout
+
+\begin_layout Proof
+Dokazati notranjost, enoto, asociativnost in inverze je trivialno.
+ Treba je še dokazati dobro definiranost, t.
+ j.
+ 
+\begin_inset Formula $a,a'$
+\end_inset
+
+ iz istega odseka in 
+\begin_inset Formula $b,b'$
+\end_inset
+
+ iz istega odseka 
+\begin_inset Formula $\Rightarrow\left(aH\right)\left(bH\right)=\left(ab\right)H=\left(a'b'\right)H=\left(a'H\right)\left(b'H\right)$
+\end_inset
+
+.
+ Ker 
+\begin_inset Formula $a'\in aH\Rightarrow\exists h'\in H\ni:ah'=a'$
+\end_inset
+
+.
+ Ker 
+\begin_inset Formula $b'\in bH\Rightarrow\exists h''\in H\ni:bh''=b'$
+\end_inset
+
+.
+ Sedaj 
+\begin_inset Formula $\left(a'H\right)\left(b'H\right)\overset{\text{def.}}{=}\left(a'b'\right)H=\left(ah'bh''\right)H=ah'b\left(h''H\right)\overset{\ref{enu:-1}}{=}ah'bH=ah'\left(bH\right)\overset{H\text{ edinka}}{=}ah'\left(Hb\right)=a\left(h'H\right)b\overset{\ref{enu:-1}}{=}aHb=\left(ab\right)H$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Kaj je kolobar, kaj je cel kolobar? Kaj je pravilo krajšanja v kolobarjih
+ in kakšna je povezava tega pravila s celimi kolobarji? Kaj velja za končne
+ cele kolobarje?
+\end_layout
+
+\begin_layout Definition*
+Bigrupoid 
+\begin_inset Formula $\left(R,+,\cdot\right)$
+\end_inset
+
+ je kolobar, če je 
+\begin_inset Formula $\left(R,+\right)$
+\end_inset
+
+ abelova grupa, 
+\begin_inset Formula $\left(R,\cdot\right)$
+\end_inset
+
+ polgrupa in velja 
+\begin_inset Formula $\forall a,b,c\in R:a\cdot\left(b+c\right)=a\cdot b+a\cdot c\wedge\left(a+b\right)\cdot c=a\cdot c+b\cdot c$
+\end_inset
+
+ (leva in desna distributivnost).
+ Kolobar je komutativen, če je 
+\begin_inset Formula $\left(R,\cdot\right)$
+\end_inset
+
+ komutativna polgrupa.
+ 
+\begin_inset Formula $\left(R,+,\cdot\right)$
+\end_inset
+
+ je kolobar z enoto, če je 
+\begin_inset Formula $\left(R,\cdot\right)$
+\end_inset
+
+ monoid.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Direktna vsota kolobarjev: Naj bosta 
+\begin_inset Formula $R$
+\end_inset
+
+ in 
+\begin_inset Formula $S$
+\end_inset
+
+ kolobarja.
+ 
+\begin_inset Formula $R\oplus S=R\times S$
+\end_inset
+
+ je direktna vsota .
+ Definirajmo operaciji 
+\begin_inset Formula $\left(r,s\right)+'\left(r',s'\right)\coloneqq\left(r+r',s+s'\right)$
+\end_inset
+
+, 
+\begin_inset Formula $\left(r,s\right)\cdot'\left(r',s'\right)\coloneqq\left(r\cdot r',s\cdot s'\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Center kolobarja 
+\begin_inset Formula $\sim ZR\coloneqq\left\{ a\in R;\forall x\in R:ax=xa\right\} $
+\end_inset
+
+, ZDB vsi taki elementi 
+\begin_inset Formula $R$
+\end_inset
+
+, ki pri množenju s poljubnim elementom kolobarja komutirajo.
+\end_layout
+
+\begin_layout Claim*
+Če sta 
+\begin_inset Formula $R$
+\end_inset
+
+ in 
+\begin_inset Formula $S$
+\end_inset
+
+ kolobarja, je 
+\begin_inset Formula $\left(R\oplus S,+',\cdot'\right)$
+\end_inset
+
+ kolobar.
+ Nadalje: Če sta 
+\begin_inset Formula $R$
+\end_inset
+
+ in 
+\begin_inset Formula $S$
+\end_inset
+
+ komutativna kolobara, je tudi 
+\begin_inset Formula $\left(R\oplus S,+',\cdot'\right)$
+\end_inset
+
+ komutativen kolobar.
+ Če sta z enoto, je tudi 
+\begin_inset Formula $\left(R\oplus S,+',\cdot'\right)$
+\end_inset
+
+.
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+DEFINIRAJ PODKOLOBAR
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Če v kolobarju 
+\begin_inset Formula $R$
+\end_inset
+
+ velja 
+\begin_inset Formula $a\cdot b=0$
+\end_inset
+
+
+\begin_inset Foot
+status open
+
+\begin_layout Plain Layout
+0 je aditivna enota
+\end_layout
+
+\end_inset
+
+ in sta 
+\begin_inset Formula $a\not=0$
+\end_inset
+
+ in 
+\begin_inset Formula $b\not=0$
+\end_inset
+
+, sta 
+\begin_inset Formula $a$
+\end_inset
+
+ in 
+\begin_inset Formula $b$
+\end_inset
+
+ delitelja niča.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+V kolobarju 
+\begin_inset Formula $\left(R,+,\cdot\right)$
+\end_inset
+
+ velja pravilo krajšanja, če velja 
+\begin_inset Formula $\forall a,b,c\in R,a\not=0:ab=ac\Rightarrow b=c$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Separator plain
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Komutativen kolobar z enoto 
+\begin_inset Formula $1\not=0$
+\end_inset
+
+ (ZDB multiplikativna enota ni enaka aditivni) brez deliteljev niča je 
+\series bold
+cel
+\series default
+.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $\left(\mathbb{Z},+,\cdot\right)$
+\end_inset
+
+ je cel, toda 
+\begin_inset Formula $\left(\mathbb{Z}_{6},+_{6},\cdot_{6}\right)$
+\end_inset
+
+ ni cel, ker premore delitelje niča.
+\end_layout
+
+\begin_layout Theorem*
+Naj bo 
+\begin_inset Formula $\left(R,+,\cdot\right)$
+\end_inset
+
+ komutativen kolobar z enoto 
+\begin_inset Formula $1\not=0$
+\end_inset
+
+.
+ Velja 
+\begin_inset Formula $R$
+\end_inset
+
+ cel 
+\begin_inset Formula $\Leftrightarrow$
+\end_inset
+
+ v 
+\begin_inset Formula $R$
+\end_inset
+
+ velja pravilo krajšanja.
+\end_layout
+
+\begin_layout Proof
+Dokazujemo ekvivalenco:
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ Predpostavimo, da je 
+\begin_inset Formula $R$
+\end_inset
+
+ cel.
+ Predpostavimo 
+\begin_inset Formula $ab=ac$
+\end_inset
+
+ za poljuben 
+\begin_inset Formula $a\not=0$
+\end_inset
+
+ in poljubna 
+\begin_inset Formula $b,c$
+\end_inset
+
+.
+ Računamo: 
+\begin_inset Formula $ab=ac\Leftrightarrow ab-ac=0\Leftrightarrow a\left(b-c\right)=0$
+\end_inset
+
+.
+ Ker je 
+\begin_inset Formula $R$
+\end_inset
+
+ cel in 
+\begin_inset Formula $a\not=0$
+\end_inset
+
+, mora biti 
+\begin_inset Formula $b-c=0$
+\end_inset
+
+, sicer bi bila 
+\begin_inset Formula $a$
+\end_inset
+
+ in 
+\begin_inset Formula $b-c$
+\end_inset
+
+ delitelja niča, kar bi bilo v 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+ s predpostavko o celosti 
+\begin_inset Formula $R$
+\end_inset
+
+.
+ 
+\begin_inset Formula $b-c=0\Rightarrow b=c$
+\end_inset
+
+, torej 
+\begin_inset Formula $\forall a\not=0,b,c\in R:ab=ac\Rightarrow b=c\sim$
+\end_inset
+
+ velja pravilo krajšanja.
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+ Predpostavimo, da v komutativnem kolobarju z enoto 
+\begin_inset Formula $1\not=0$
+\end_inset
+
+ 
+\begin_inset Formula $R$
+\end_inset
+
+ velja pravilo krajšanja.
+ Predpostavimo 
+\begin_inset Formula $ab=0$
+\end_inset
+
+, 
+\begin_inset Formula $a\not=0$
+\end_inset
+
+.
+ Dokazati je treba 
+\begin_inset Formula $b=0$
+\end_inset
+
+, sicer bi imeli delitelje niča.
+ Velja 
+\begin_inset Formula $ab=0=a\cdot0$
+\end_inset
+
+.
+ Uporabimo pravilo krajšanja na 
+\begin_inset Formula $ab=a0$
+\end_inset
+
+ in dobimo 
+\begin_inset Formula $b=0$
+\end_inset
+
+.
+ Kolobar je cel.
+\end_layout
+
+\end_deeper
+\begin_layout Definition*
+Komutativen kolobar 
+\begin_inset Formula $R$
+\end_inset
+
+ z enoto 
+\begin_inset Formula $1\not=0$
+\end_inset
+
+ je 
+\series bold
+obseg
+\series default
+, če je vsak neničeln element obrnljiv v 
+\begin_inset Formula $\left(R,\cdot\right)$
+\end_inset
+
+ ZDB 
+\begin_inset Formula $\left(R\setminus\left\{ 0\right\} ,\cdot\right)$
+\end_inset
+
+ je grupa.
+ Obseg 
+\begin_inset Formula $R$
+\end_inset
+
+ je polje, če je 
+\begin_inset Formula $\left(R\setminus\left\{ 0\right\} ,\cdot\right)$
+\end_inset
+
+ abelova grupa ZDB 
+\begin_inset Formula $ZR=R$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Če je 
+\begin_inset Formula $\left(R,+,\cdot\right)$
+\end_inset
+
+ končen cel kolobar 
+\begin_inset Formula $\Rightarrow R$
+\end_inset
+
+ polje.
+\end_layout
+
+\begin_layout Proof
+Dokažimo, da 
+\begin_inset Formula $\forall a\in R,a\not=0\exists a^{-1}\ni:aa^{-1}=1$
+\end_inset
+
+.
+ Naj bo 
+\begin_inset Formula $a\in R$
+\end_inset
+
+ poljuben, z izjemo 
+\begin_inset Formula $a\not=0$
+\end_inset
+
+.
+ Oglejmo si 
+\begin_inset Formula $\left\{ a^{k};\forall k\geq0\right\} $
+\end_inset
+
+ ZDB množico vseh potenc 
+\begin_inset Formula $a$
+\end_inset
+
+.
+ Ker 
+\begin_inset Formula $\left|R\right|<\infty\Rightarrow\exists i,j,$
+\end_inset
+
+BSŠ 
+\begin_inset Formula $i>j\ni a^{i}=a^{j}$
+\end_inset
+
+ ZDB ker je 
+\begin_inset Formula $R$
+\end_inset
+
+ končen, se nek element 
+\begin_inset Quotes gld
+\end_inset
+
+ponovi
+\begin_inset Quotes grd
+\end_inset
+
+.
+ 
+\begin_inset Formula $a^{j}\cdot a^{i-j}=a^{i}=a^{j}=a^{j}\cdot1$
+\end_inset
+
+ Ker je kolobar cel, 
+\begin_inset Formula $a^{j}\not=0$
+\end_inset
+
+ in ker velja pravilo krajšanja, 
+\begin_inset Formula $a^{i-j}=1$
+\end_inset
+
+, pri čemer vemo, da je 
+\begin_inset Formula $i-j>0$
+\end_inset
+
+.
+ Ločimo dva primera:
+\end_layout
+
+\begin_deeper
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $i-j=1$
+\end_inset
+
+ 
+\begin_inset Formula $a=1\Rightarrow a$
+\end_inset
+
+ je kot multiplikativna enota multiplikativni inverz sam sebi
+\end_layout
+
+\begin_layout Labeling
+\labelwidthstring 00.00.0000
+\begin_inset Formula $i-j>1$
+\end_inset
+
+ 
+\begin_inset Formula $a=a\cdot a^{i-j-1}=1$
+\end_inset
+
+, torej 
+\begin_inset Formula $a^{i-j-1}$
+\end_inset
+
+ je inverz 
+\begin_inset Formula $a$
+\end_inset
+
+.
+\end_layout
+
+\end_deeper
+\begin_layout Subsection
+Kaj je karakteristika kolobarja? Kako lahko določimo karakteristiko kolobarja
+ z enoto? Kaj lahko povemo o karakteristiki celega kolobarja?
+\end_layout
+
+\begin_layout Definition*
+Karakteristika kolobarja 
+\begin_inset Formula $R\sim\karakteristika R$
+\end_inset
+
+ je najmanjši 
+\begin_inset Formula $n\in\mathbb{N}\ni:\forall a\in R:a+\cdots_{n-\text{krat}}\cdots+a=0$
+\end_inset
+
+.
+ Če tak 
+\begin_inset Formula $n$
+\end_inset
+
+ ne obstaja, pravimo 
+\begin_inset Formula $\karakteristika R=0$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Naj bo 
+\begin_inset Formula $\left(R,+,\cdot\right)$
+\end_inset
+
+ kolobar z enoto.
+ 
+\begin_inset Formula $\karakteristika R=\red1$
+\end_inset
+
+ v grupi 
+\begin_inset Formula $\left(R,+\right)$
+\end_inset
+
+ ZDB red enote v aditivni grupi.
+\end_layout
+
+\begin_layout Proof
+Po definiciji reda je 
+\begin_inset Formula $1+\cdots_{\red1-\text{krat}}\cdots+1=0$
+\end_inset
+
+ in za nek 
+\begin_inset Formula $m<\red1$
+\end_inset
+
+ 
+\begin_inset Formula $1+\cdots_{m-\text{krat}}\cdots+1\not=0$
+\end_inset
+
+, torej 
+\begin_inset Formula $\karakteristika R\geq\red1$
+\end_inset
+
+.
+ Sedaj vzemimo poljuben 
+\begin_inset Formula $a\in R$
+\end_inset
+
+.
+ Velja 
+\begin_inset Formula $a+\cdots_{\red1-\text{krat}}\cdots+a=1\cdot a+\cdots_{\red1-\text{krat}}\cdots+1\cdot a=a\cdot\left(1+\cdots_{\red1-\text{krat}}\cdots+1\right)=a\cdot0=0$
+\end_inset
+
+, torej 
+\begin_inset Formula $\karakteristika R\leq\red1$
+\end_inset
+
+, torej 
+\begin_inset Formula $\karakteristika R=\red1$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Naj bo 
+\begin_inset Formula $\left(R,+,\cdot\right)$
+\end_inset
+
+ cel kolobar.
+ Tedaj velja bodisi 
+\begin_inset Formula $\karakteristika R=0$
+\end_inset
+
+ bodisi 
+\begin_inset Formula $\karakteristika R=p$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $p$
+\end_inset
+
+ praštevilo.
+\end_layout
+
+\begin_layout Proof
+Če je 
+\begin_inset Formula $\karakteristika R=0$
+\end_inset
+
+, smo dokazali, sicer je 
+\begin_inset Formula $\karakteristika R=r>0$
+\end_inset
+
+.
+ PDDRAA 
+\begin_inset Formula $n=pq$
+\end_inset
+
+ za 
+\begin_inset Formula $p,q>1$
+\end_inset
+
+ in 
+\begin_inset Formula $p,q\in\mathbb{N}$
+\end_inset
+
+.
+ Po prejšnjem izreku 
+\begin_inset Formula $\karakteristika R=\red1=n$
+\end_inset
+
+.
+ Tedaj velja 
+\begin_inset Formula $0=1+\cdots_{n-\text{krat}}\cdots+1=1+\cdots_{pq-\text{krat}}\cdots+1=\left(1+\cdots_{p-\text{krat}}\cdots+1\right)\cdot\left(1+\cdots_{q-\text{krat}}\cdots+1\right)=a\cdot b$
+\end_inset
+
+.
+ Ker smo v celem kolobarju, je bodisi 
+\begin_inset Formula $a=0$
+\end_inset
+
+, bodisi 
+\begin_inset Formula $b=0$
+\end_inset
+
+, toda to je 
+\begin_inset Formula $\rightarrow\!\leftarrow$
+\end_inset
+
+, saj bi bil potem 
+\begin_inset Formula $\red1=q$
+\end_inset
+
+ ali 
+\begin_inset Formula $\red1=p$
+\end_inset
+
+, oboje pa je 
+\begin_inset Formula $<pq$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Subsection
+Kaj je ideal kolobarja? Kako lahko preverimo, ali je 
+\begin_inset Formula $I\subseteq R$
+\end_inset
+
+ ideal kolobarja 
+\begin_inset Formula $R$
+\end_inset
+
+? Kaj je faktorski kolobar 
+\begin_inset Formula $R/I$
+\end_inset
+
+ in kako računamo v njem?
+\end_layout
+
+\begin_layout Definition*
+Če je 
+\begin_inset Formula $R$
+\end_inset
+
+ kolobar, je njegov podkolobar 
+\begin_inset Formula $S$
+\end_inset
+
+ ideal, če velja 
+\begin_inset Formula $\forall v\in R,s\in S:rs,sr\in S$
+\end_inset
+
+ ZDB to je podkolobar, zaprt za zunanje množenje.
+\end_layout
+
+\begin_layout Example*
+\begin_inset Formula $n\cdot\mathbb{Z}$
+\end_inset
+
+ (večkratniki 
+\begin_inset Formula $n$
+\end_inset
+
+) za fiksen 
+\begin_inset Formula $n$
+\end_inset
+
+ so ideal v 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\mathbb{Z}$
+\end_inset
+
+ je podkolobar v 
+\begin_inset Formula $\mathbb{Q}$
+\end_inset
+
+, toda ni njegov ideal.
+\end_layout
+
+\begin_layout Claim*
+\begin_inset Formula $I\subseteq R$
+\end_inset
+
+ je ideal 
+\begin_inset Formula $\Leftrightarrow0\in I$
+\end_inset
+
+ (vsebuje aditivno enoto) 
+\begin_inset Formula $\wedge\forall i,j\in I:i-j\in I$
+\end_inset
+
+ (zaprt za odštevanje) 
+\begin_inset Formula $\wedge\forall i\in I,r\in R:ir\in I,ri\in I$
+\end_inset
+
+ (zaprt za zunanje množenje)
+\begin_inset Note Note
+status open
+
+\begin_layout Plain Layout
+TODO XXX FIXME vsota in produkt idealov je spet ideal
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Definition*
+Naj bo 
+\begin_inset Formula $R$
+\end_inset
+
+ kolobar in 
+\begin_inset Formula $I$
+\end_inset
+
+ ideal v 
+\begin_inset Formula $R$
+\end_inset
+
+.
+ Faktorski kolobar: 
+\begin_inset Formula $R/I=\left\{ a+I,\forall a\in R\right\} =\left\{ \left\{ a+i;\forall i\in I\right\} ;\forall a\in R\right\} $
+\end_inset
+
+ vsebuje aditivne odseke.
+ V 
+\begin_inset Formula $R/I$
+\end_inset
+
+ vpeljemo operaciji: 
+\begin_inset Formula $\left(a+I\right)+'\left(b+I\right)=a+b+I$
+\end_inset
+
+ in 
+\begin_inset Formula $\left(a+I\right)\cdot'\left(b+I\right)=a\cdot b+I$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Theorem*
+Če je 
+\begin_inset Formula $I$
+\end_inset
+
+ ideal v 
+\begin_inset Formula $R$
+\end_inset
+
+, je 
+\begin_inset Formula $\left(R/I,+',\cdot'\right)$
+\end_inset
+
+ kolobar.
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/krožek/05-17.odp b/šola/krožek/05-17.odp
new file mode 100644
index 0000000..244b2b0
--- /dev/null
+++ b/šola/krožek/05-17.odp
diff --git a/šola/krožek/funkcije.odp b/šola/krožek/funkcije.odp
new file mode 100644
index 0000000..fe98438
--- /dev/null
+++ b/šola/krožek/funkcije.odp
diff --git a/šola/la/dn8/dokument.lyx b/šola/la/dn8/dokument.lyx
new file mode 100644
index 0000000..c603fcc
--- /dev/null
+++ b/šola/la/dn8/dokument.lyx
@@ -0,0 +1,1349 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
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+\use_hyperref false
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+\use_package amsmath 1
+\use_package amssymb 1
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+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 2cm
+\topmargin 2cm
+\rightmargin 2cm
+\bottommargin 2cm
+\headheight 2cm
+\headsep 2cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
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+\quotes_style german
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+\paperpagestyle default
+\tracking_changes false
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+\html_math_output 0
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+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Title
+Rešitev osme domače naloge Linearne Algebre
+\end_layout
+
+\begin_layout Author
+
+\noun on
+Anton Luka Šijanec
+\end_layout
+
+\begin_layout Date
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+today
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Abstract
+Za boljšo preglednost sem svoje rešitve domače naloge prepisal na računalnik.
+ Dokumentu sledi še rokopis.
+ Naloge je izdelala asistentka Ajda Lemut.
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+newcommand
+\backslash
+euler{e}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Enumerate
+Dokaži, da je 
+\begin_inset Formula $\left[\left(x,y,z\right),\left(u,v,w\right)\right]=2xu-yu-xv+2yv-zv-yw+zw$
+\end_inset
+
+ skalarni produkt in ugotovi, ali je
+\begin_inset Formula 
+\[
+A=\left[\begin{array}{ccc}
+0 & 2 & -2\\
+0 & 1 & 0\\
+-1 & 2 & -1
+\end{array}\right]
+\]
+
+\end_inset
+
+ normalna preslikava glede na 
+\begin_inset Formula $\left[\cdot,\cdot\right]$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Standard
+Predpostavljam polje 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ in vektorski prostor 
+\begin_inset Formula $V=\mathbb{R}^{3}$
+\end_inset
+
+, saj v kompleksnem to ni skalarni produkt (protiprimer pozitivne definitnosti
+ je 
+\begin_inset Formula $\left[\left(1,1,1+i\right),\left(1,1,1+i\right)\right]=2$
+\end_inset
+
+).
+ 
+\begin_inset Formula $\langle\cdot,\cdot\rangle:V\times V\to\mathbb{R}$
+\end_inset
+
+ je skalarni produkt, če zadošča naslednjim lastnostim.
+ Dokažimo jih za 
+\begin_inset Formula $\left[\cdot,\cdot\right]$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall v\in V:v\not=0\Rightarrow\langle v,v\rangle>0$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left[\left(x,y,z\right),\left(x,y,z\right)\right]=2x^{2}-2xy+2y^{2}-2yz+z^{2}=2\left(x^{2}-xy+y^{2}\right)-2yz+z^{2}=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=2\left(\left(x-\frac{y}{2}\right)^{2}+\frac{3y^{2}}{4}\right)-2yz+z^{2}=2\left(x-\frac{y}{2}\right)^{2}-\frac{3y^{2}}{4}-2yz+z^{2}=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=2\left(x-\frac{y}{2}\right)^{2}+\left(\frac{\sqrt{3}y}{\sqrt{2}}-\frac{z\sqrt{2}}{\sqrt{3}}\right)^{2}+\frac{z^{2}}{3}\geq0
+\]
+
+\end_inset
+
+Sedaj poiščimo ničle.
+ Fiksirajmo poljubna 
+\begin_inset Formula $y$
+\end_inset
+
+, 
+\begin_inset Formula $z$
+\end_inset
+
+ in uporabimo obrazec za ničle kvadratne enačbe:
+\begin_inset Formula 
+\[
+x_{1,2}=\frac{2y\pm\sqrt{4y^{2}-8\left(2y^{2}-2yz+z^{2}\right)}}{4}
+\]
+
+\end_inset
+
+Iščemo pozitivne diskriminante.
+\begin_inset Formula 
+\[
+4y^{2}-8\left(2y^{2}-2yz+z^{2}\right)=-12y^{2}+16yz-8z^{2}=4
+\]
+
+\end_inset
+
+Fiksirajmo poljuben 
+\begin_inset Formula $z$
+\end_inset
+
+.
+ Vodilni koeficient kvadratne enačbe je negativen.
+ Uporabimo obrazec:
+\begin_inset Formula 
+\[
+y_{1,2}=\frac{-16z\pm\sqrt{256z^{2}-384z^{2}=-128z^{2}}}{-24}
+\]
+
+\end_inset
+
+Diskriminanta je nenegativna 
+\begin_inset Formula $\Leftrightarrow z=0$
+\end_inset
+
+.
+ Torej 
+\begin_inset Formula $z=0$
+\end_inset
+
+, zato 
+\begin_inset Formula $y=0$
+\end_inset
+
+ in tudi 
+\begin_inset Formula $x=0$
+\end_inset
+
+ glede na obrazce.
+ Skalarni produkt je res pozitivno definiten.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall v,u\in V:\langle v,u\rangle=\langle u,v\rangle$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left[\left(u,v,w\right),\left(x,y,z\right)\right]=\left[\left(x,y,z\right),\left(u,v,w\right)\right]
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+2ux-vx-uy-2vy-wy-vz+wz=2xu-yu-xv+2yv-zv-vz+wz
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+0=0
+\]
+
+\end_inset
+
+Skalarni produkt je res simetričen.
+\end_layout
+
+\begin_layout Enumerate
+\begin_inset Formula $\forall\alpha_{1},\alpha_{2}\in\mathbb{C}\forall u_{1},u_{2},v\in V:\langle\alpha_{1}v_{1}+\alpha_{2}v_{2},v\rangle=\alpha_{1}\langle u_{1},v\rangle+\alpha_{2}\langle u_{2},v\rangle$
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left[\alpha_{1}\left(x_{1},y_{1},z_{1}\right)+\alpha_{2}\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right]=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\left[\left(\alpha_{1}x_{1}+\alpha_{2}x_{2},\alpha_{1}y_{1}+\alpha_{2}y_{2},\alpha_{1}z_{1}+\alpha_{2}z_{2}\right),\left(u,v,w\right)\right]=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=2\left(\alpha_{1}x_{1}+\alpha_{2}x_{2}\right)u-\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)u-\left(\alpha_{1}x_{1}+\alpha_{2}x_{2}\right)v+
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
++2\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)v-\left(\alpha_{1}z_{1}+\alpha_{2}z_{2}\right)v-\left(\alpha_{1}y_{1}+\alpha_{2}y_{2}\right)w+\left(\alpha_{1}z_{1}+\alpha_{2}z_{2}\right)w=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=2\alpha_{1}x_{1}u+2\alpha_{2}x_{2}u-\alpha_{1}y_{1}u-\alpha_{2}y_{2}u-\alpha_{1}x_{1}v-\alpha_{2}x_{2}v+
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
++2\alpha_{1}y_{1}v+2\alpha_{2}y_{2}v-\alpha_{1}z_{1}v-\alpha_{2}z_{2}v-\alpha_{1}y_{1}w-\alpha_{2}y_{2}w+\alpha_{1}z_{1}w+\alpha_{2}z_{2}w=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\alpha_{1}\left(2x_{1}u-y_{1}u-x_{1}v+2y_{1}v-z_{1}v-y_{1}w+z_{1}w\right)+\alpha_{2}\left(2x_{2}u-y_{2}u-x_{2}v+2y_{2}v-z_{2}v-y_{2}w+z_{2}w\right)=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\alpha_{1}\left[\left(x_{1},y_{1},z_{1}\right),\left(u,v,w\right)\right]+\alpha_{2}\left[\left(x_{2},y_{2},z_{2}\right),\left(u,v,w\right)\right]
+\]
+
+\end_inset
+
+Skalarni produkt je res homogen in linearen v prvem faktorju.
+\end_layout
+
+\begin_layout Standard
+Po definiciji 
+\begin_inset Formula $A$
+\end_inset
+
+ normalna 
+\begin_inset Formula $\Leftrightarrow A^{*}A=AA^{*}$
+\end_inset
+
+.
+ Izračunajmo torej matriko 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Na predavanjih 2024-05-08 smo dokazali, da za vsak skalarni produkt 
+\begin_inset Formula $\left[u,v\right]$
+\end_inset
+
+ obstaja taka ortogonalna (
+\begin_inset Formula $M^{*}=M^{-1}$
+\end_inset
+
+) pozitivno definitna matrika 
+\begin_inset Formula $M$
+\end_inset
+
+, da velja 
+\begin_inset Formula $\left[u,v\right]=\langle u,Mv\rangle$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $\langle\cdot,\cdot\rangle$
+\end_inset
+
+ standardni skalarni produkt.
+\end_layout
+
+\begin_layout Itemize
+Izpeljimo predpis za 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ pri podani matriki 
+\begin_inset Formula $A$
+\end_inset
+
+ in skalarnem produktu 
+\begin_inset Formula $\left[\cdot,\cdot\right]$
+\end_inset
+
+ s pripadajočo matriko 
+\begin_inset Formula $M$
+\end_inset
+
+:
+\begin_inset Formula 
+\[
+\left[A^{*}x,y\right]=\left[x,Ay\right]\text{, uporabimo prvo točko:}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left\langle A^{*}x,My\right\rangle =\left\langle x,MAy\right\rangle \text{, pišimo \ensuremath{z=My}:}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left\langle A^{*}x,z\right\rangle =\left\langle x,MAM^{-1}z\right\rangle \text{, upoštevajmo drugo točko:}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left\langle A^{*}x,z\right\rangle =\left\langle M^{-1}A^{\square}Mx,z\right\rangle \text{, kjer je \ensuremath{A^{\square}} adjungacija \ensuremath{A} pri standardnem skalarnem produktu}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\Rightarrow A^{*}=M^{-1}A^{\square}M=M^{-1}\overline{A}^{T}M\overset{A\in M\left(\mathbb{R}\right)}{=}M^{-1}A^{T}M
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Potrebujemo še matriko skalarnega produkta.
+\begin_inset Formula 
+\[
+\left\langle \left(x,y,z\right),M\left(u,v,w\right)\right\rangle =\left[\begin{array}{ccc}
+x & y & z\end{array}\right]\left[\begin{array}{ccc}
+m_{11} & m_{12} & m_{13}\\
+m_{21} & m_{22} & m_{23}\\
+m_{31} & m_{32} & m_{33}
+\end{array}\right]\left[\begin{array}{c}
+u\\
+v\\
+w
+\end{array}\right]=\left[\begin{array}{ccc}
+x & y & z\end{array}\right]\left[\begin{array}{c}
+um_{11}+vm_{12}+wm_{13}\\
+um_{21}+vm_{22}+wm_{23}\\
+um_{31}+vm_{32}+wm_{33}
+\end{array}\right]=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\begin{array}{ccccc}
+ & xum_{11} & xvm_{12} & xwm_{13} & +\\
++ & yum_{21} & yvm_{22} & ywm_{23} & +\\
++ & zum_{31} & zvm_{32} & zwm_{33}
+\end{array}=\left[\left(x,y,z\right),\left(u,v,w\right)\right]=2xu-yu-xv+2yv-zv-yw+zw\text{, torej}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+M=\left[\begin{array}{ccc}
+2 & -1 & 0\\
+-1 & 2 & -1\\
+0 & -1 & 1
+\end{array}\right]\text{, njen inverz pa je }M^{-1}=\left[\begin{array}{ccc}
+1 & 1 & 1\\
+1 & 2 & 2\\
+1 & 2 & 3
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Izračunamo 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ po formuli 
+\begin_inset Formula $A^{*}=M^{-1}A^{T}M$
+\end_inset
+
+ in preverimo 
+\begin_inset Formula $A^{*}A=AA^{*}$
+\end_inset
+
+:
+\begin_inset Formula 
+\[
+A^{*}=\left[\begin{array}{ccc}
+1 & 1 & 1\\
+1 & 2 & 2\\
+1 & 2 & 3
+\end{array}\right]\left[\begin{array}{ccc}
+0 & 0 & -1\\
+2 & 1 & 2\\
+-2 & 0 & -1
+\end{array}\right]\left[\begin{array}{ccc}
+2 & -1 & 0\\
+-1 & 2 & -1\\
+0 & -1 & 1
+\end{array}\right]=\left[\begin{array}{ccc}
+-1 & 2 & -1\\
+-2 & 3 & -1\\
+-6 & 6 & -2
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+A^{*}A=\left[\begin{array}{ccc}
+1 & -2 & 3\\
+1 & -3 & 5\\
+2 & -10 & 14
+\end{array}\right]\not=\left[\begin{array}{ccc}
+8 & -6 & 2\\
+-2 & 3 & -1\\
+3 & -2 & 1
+\end{array}\right]=AA^{*}
+\]
+
+\end_inset
+
+
+\begin_inset Formula $A$
+\end_inset
+
+ ni normala matrika.
+\end_layout
+
+\begin_layout Itemize
+Da preverimo pravilnost matrike 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+, lahko napravimo preizkus:
+\begin_inset Float figure
+placement H
+wide false
+sideways false
+status open
+
+\begin_layout Plain Layout
+\begin_inset Graphics
+	filename sage.png
+	width 100col%
+
+\end_inset
+
+
+\begin_inset Caption Standard
+
+\begin_layout Plain Layout
+Preizkus s programom SageMath.
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Dokazati, da 
+\begin_inset Formula $A$
+\end_inset
+
+ ni normalna, je moč še lažje.
+ Dokažemo lahko namreč, da eden izmed potrebnih pogojev za normalnost matrike
+ ni izpolnjen.
+ Na primer: 
+\begin_inset Formula $AA^{*}=A^{*}A\rightarrow A=PDP^{-1}$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $P$
+\end_inset
+
+ ortogonalna in 
+\begin_inset Formula $D$
+\end_inset
+
+ diagonalna 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ lastni vektorji 
+\begin_inset Formula $A$
+\end_inset
+
+ tvorijo ortogonalno množico.
+\end_layout
+
+\begin_layout Standard
+Lastne vrednosti 
+\begin_inset Formula $A$
+\end_inset
+
+ so (s kalkulatorjem) 
+\begin_inset Formula $\left\{ -2,1\right\} $
+\end_inset
+
+, kjer ima 1 algebrajsko večkratnost 2.
+ Lastni vektorji:
+\begin_inset Formula 
+\[
+A-\left(-2\right)I=\left[\begin{array}{ccc}
+2 & 2 & -2\\
+0 & 3 & 0\\
+-1 & 2 & 1
+\end{array}\right]\sim\left[\begin{array}{ccc}
+2 & 2 & -2\\
+0 & 3 & 0\\
+0 & 3 & 0
+\end{array}\right]\sim\left[\begin{array}{ccc}
+2 & 2 & -2\\
+0 & 3 & 0\\
+0 & 0 & 0
+\end{array}\right]\sim\left[\begin{array}{ccc}
+2 & 0 & -2\\
+0 & 3 & 0\\
+0 & 0 & 0
+\end{array}\right]\Rightarrow x=z,y=0\Rightarrow v_{1}=\left(1,0,1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+A-1I=\left[\begin{array}{ccc}
+-1 & 2 & -2\\
+0 & 0 & 0\\
+-1 & 2 & -2
+\end{array}\right]\sim\left[\begin{array}{ccc}
+-1 & 2 & -2\\
+0 & 0 & 0\\
+0 & 0 & 0
+\end{array}\right]\Rightarrow x=2y-2z\Rightarrow v_{2}=\left(2,1,0\right),\quad v_{3}=\left(-2,0,1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left[v_{1},v_{2}\right]=\left[\left(1,0,1\right),\left(2,1,0\right)\right]=4-0-1+0-1-0+0=2\not=0\Rightarrow v_{1}\not\perp v_{2}\Rightarrow A\text{ ni normalna}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Pokaži 
+\begin_inset Formula $A:V\to V$
+\end_inset
+
+ je normalna 
+\begin_inset Formula $\Leftrightarrow AA^{*}-A^{*}A$
+\end_inset
+
+ je pozitivno semidefinitna.
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Itemize
+Definiciji:
+\end_layout
+
+\begin_deeper
+\begin_layout Itemize
+\begin_inset Formula $A:V\to V$
+\end_inset
+
+ je normalna 
+\begin_inset Formula $\Leftrightarrow A^{*}A=AA^{*}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $A:V\to V$
+\end_inset
+
+ je pozitivno semidefinitna 
+\begin_inset Formula $\Leftrightarrow A=A^{*}\wedge\forall v\in V:\left\langle Av,v\right\rangle \geq0$
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Itemize
+\begin_inset Formula $\left(\Rightarrow\right)$
+\end_inset
+
+ Po predpostavki velja 
+\begin_inset Formula $AA^{*}=A^{*}A\Rightarrow AA^{*}-A^{*}A=0$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+AA^{*}-A^{*}A\overset{?}{=}\left(AA^{*}-A^{*}A\right)^{*}\Leftrightarrow0=0^{*}
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left\langle \left(AA^{*}-A^{*}A\right)v,v\right\rangle =\left\langle 0v,v\right\rangle \overset{\text{homogenost}}{=}0\left\langle v,v\right\rangle =0\geq0
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+\begin_inset Formula $\left(\Leftarrow\right)$
+\end_inset
+
+Po predpostavki velja 
+\begin_inset Formula $\left(AA^{*}-A^{*}A\right)^{*}=AA^{*}-A^{*}A$
+\end_inset
+
+ in 
+\begin_inset Formula $\forall v\in V:\left\langle \left(AA^{*}-A^{*}A\right)v,v\right\rangle \geq0$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+\sled\left(AA^{*}-A^{*}A\right)=\sled\left(AA^{*}\right)-\sled\left(A^{*}A\right)\overset{\text{lastnost sledi}}{=}\sled\left(AA^{*}\right)-\sled\left(A^{*}A\right)=0
+\]
+
+\end_inset
+
+Sled 
+\begin_inset Formula $M$
+\end_inset
+
+ je vsota lastnih vrednosti 
+\begin_inset Formula $M$
+\end_inset
+
+, torej je vsota lastnih vrednosti 
+\begin_inset Formula $\left(AA^{*}-A^{*}A\right)=0$
+\end_inset
+
+.
+ 
+\begin_inset Formula $AA^{*}-A^{*}A\geq0\Rightarrow$
+\end_inset
+
+ vse lastne vrednosti so nenegativne.
+ Iz teh dveh trditev sledi, da je vsaka lastna vrednost 
+\begin_inset Formula $AA^{*}-A^{*}A=0$
+\end_inset
+
+.
+ 
+\begin_inset Formula $AA^{*}-A^{*}A\geq0\Rightarrow AA^{*}-A^{*}A$
+\end_inset
+
+ normalna.
+ Normalne matrike je moč diagonalizirati v ortonormirani bazi:
+\begin_inset Formula 
+\[
+AA^{*}-A^{*}A=PDP^{-1}\overset{\text{diagonalci so lastne vrednosti}}{=}P0P^{-1}=0
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+AA^{*}=A^{*}A\Rightarrow A\text{ je normalna}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Naj bo 
+\begin_inset Formula $w_{1}=\left(1,1,1,1\right)$
+\end_inset
+
+, 
+\begin_inset Formula $w_{2}=\left(3,3,-1,-1\right)$
+\end_inset
+
+ in 
+\begin_inset Formula $y=\left(6,0,2,0\right)$
+\end_inset
+
+.
+\end_layout
+
+\begin_deeper
+\begin_layout Enumerate
+Poišči ortonormirano bazo za 
+\begin_inset Formula $W=\Lin\left\{ w_{1},w_{2}\right\} $
+\end_inset
+
+ glede na standardni skalarni produkt.
+\end_layout
+
+\begin_layout Enumerate
+Izrazi 
+\begin_inset Formula $y$
+\end_inset
+
+ kot vsoto vektorja iz 
+\begin_inset Formula $W$
+\end_inset
+
+ in vektorja iz 
+\begin_inset Formula $W^{\perp}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Enumerate
+Uporabimo Gram-Schmidtov postopek in sproti normiramo bazne vektorje:
+\begin_inset Formula 
+\[
+v_{1}=\left(1,1,1,1\right)/2=\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\tilde{v_{2}}=\left(3,3,-1,-1\right)-\left\langle \left(3,3,-1,-1\right),v_{1}\right\rangle v_{1}=\left(3,3,-1,-1\right)-\left\langle \left(3,3,-1,-1\right),\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right\rangle \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\left(3,3,-1,-1\right)-2\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)=\left(2,2,-2,-2\right),\quad\quad\quad v_{2}=\tilde{v_{2}}/\left|\left|\tilde{v_{2}}\right|\right|=\text{\ensuremath{\tilde{v_{2}}/4}}=\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)
+\]
+
+\end_inset
+
+Baza za 
+\begin_inset Formula $W$
+\end_inset
+
+ je 
+\begin_inset Formula $B=\left\{ v_{1},v_{2}\right\} =\left\{ \left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\} $
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Enumerate
+Dopolnimo 
+\begin_inset Formula $B$
+\end_inset
+
+ do baze 
+\begin_inset Formula $\mathbb{R}^{4}$
+\end_inset
+
+.
+ Dopolnitev 
+\begin_inset Formula $\left\{ u_{1},u_{2}\right\} $
+\end_inset
+
+ bo ortonormirana baza za 
+\begin_inset Formula $W^{\perp}$
+\end_inset
+
+, nato uporabimo Fourierov razvoj po dopolnjeni bazi.
+ Bazo podprostora dopolnimo tako, da rešimo sistem enačb.
+\begin_inset Formula 
+\[
+\left\langle \left(x_{1},y_{1},z_{1},w_{1}\right),\left(3,3,-1,-1\right)\right\rangle =0\quad\quad\quad\left\langle \left(x_{2},y_{2},z_{2},w_{2}\right),\left(1,1,1,1\right)\right\rangle =0
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\left[\begin{array}{cccc}
+1 & 1 & 1 & 1\\
+3 & 3 & -1 & -1
+\end{array}\right]\sim\left[\begin{array}{cccc}
+1 & 1 & 0 & 0\\
+0 & 0 & 1 & 1
+\end{array}\right]\Rightarrow x=-y,\quad z=-w\Rightarrow\tilde{u_{1}}=\left(1,-1,0,0\right),\quad\tilde{u_{2}}=\left(0,0,1,-1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+u_{1}=\tilde{u_{1}}/\left|\left|\tilde{u_{1}}\right|\right|=\left(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0,0\right)\quad\quad\quad u_{2}=\tilde{u_{2}}/\left|\left|\tilde{u}_{2}\right|\right|=\left(0,0,\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}}\right)
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula 
+\[
+y=\sum_{i=1}^{n}\frac{\left\langle y,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}=\left\langle \left(6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\right\rangle v_{1}+\left\langle \left(6,0,2,0\right),\left(\frac{1}{2},\frac{1}{2},\frac{-1}{2},\frac{-1}{2}\right)\right\rangle v_{2}+
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
++\left\langle \left(6,0,2,0\right),\left(\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},0,0\right)\right\rangle u_{1}+\left\langle \left(6,0,2,0\right),\left(0,0,\frac{1}{\sqrt{2}},\frac{-1}{\sqrt{2}},\right)\right\rangle u_{2}=4v_{1}+2v_{2}+\frac{6}{\sqrt{2}}u_{1}+\frac{2}{\sqrt{2}}u_{2}=
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+=\left(2,2,2,2\right)+\left(1,1,-1,-1\right)+\left(3,-3,0,0\right)+\left(0,0,1,-1\right)=\left(3,3,1,1\right)\in W+\left(3,-3,1,-1\right)\in W^{\perp}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Enumerate
+Poišči singularni razcep matrike
+\begin_inset Formula 
+\[
+A=\left[\begin{array}{ccc}
+1 & 0 & 0\\
+0 & -2 & 0\\
+0 & 0 & 0\\
+0 & 0 & 0
+\end{array}\right]\text{.}
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_deeper
+\begin_layout Paragraph
+Rešitev
+\end_layout
+
+\begin_layout Itemize
+Iščemo 
+\begin_inset Formula $U$
+\end_inset
+
+, 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+ in 
+\begin_inset Formula $V$
+\end_inset
+
+, da velja 
+\begin_inset Formula $A=U\Sigma V^{*}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Itemize
+Diagonalci 
+\begin_inset Formula $\Sigma$
+\end_inset
+
+ so singularne vrednosti 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Singularne vrednosti 
+\begin_inset Formula $A$
+\end_inset
+
+ so koreni lastnih vrednosti 
+\begin_inset Formula $A^{*}A$
+\end_inset
+
+, torej 
+\begin_inset Formula $\sigma_{1}=2$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{2}=1$
+\end_inset
+
+, 
+\begin_inset Formula $\sigma_{3}=0$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+A^{*}A=\left[\begin{array}{cccc}
+1 & 0 & 0 & 0\\
+0 & -2 & 0 & 0\\
+0 & 0 & 0 & 0
+\end{array}\right]\left[\begin{array}{ccc}
+1 & 0 & 0\\
+0 & -2 & 0\\
+0 & 0 & 0\\
+0 & 0 & 0
+\end{array}\right]=\left[\begin{array}{ccc}
+1 & 0 & 0\\
+0 & 4 & 0\\
+0 & 0 & 0
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+\Sigma=\left[\begin{array}{ccc}
+\sigma_{1} & 0 & 0\\
+0 & \sigma_{2} & 0\\
+0 & 0 & \sigma_{3}\\
+0 & 0 & 0
+\end{array}\right]=\left[\begin{array}{ccc}
+1 & 0 & 0\\
+0 & 2 & 0\\
+0 & 0 & 0\\
+0 & 0 & 0
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Stolpci 
+\begin_inset Formula $V$
+\end_inset
+
+ so ortonormirana baza jedra 
+\begin_inset Formula $A^{*}A-\sigma^{2}I$
+\end_inset
+
+ za vse singularne vrednosti 
+\begin_inset Formula $\sigma$
+\end_inset
+
+.
+\begin_inset Formula 
+\[
+A^{*}A-4I=\left[\begin{array}{ccc}
+-3 & 0 & 0\\
+0 & 0 & 0\\
+0 & 0 & -4
+\end{array}\right]\Rightarrow x=z=0\Rightarrow v_{1}=\left(0,1,0\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+A^{*}A-1I=\left[\begin{array}{ccc}
+0 & 0 & 0\\
+0 & 3 & 0\\
+0 & 0 & -1
+\end{array}\right]\Rightarrow y=z=0\Rightarrow v_{2}=\left(1,0,0\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+A^{*}A-0I=\left[\begin{array}{ccc}
+1 & 0 & 0\\
+0 & 4 & 0\\
+0 & 0 & 0
+\end{array}\right]\Rightarrow x=y=0\Rightarrow v_{3}=\left(0,0,1\right)
+\]
+
+\end_inset
+
+
+\begin_inset Formula 
+\[
+V=\left[\begin{array}{ccc}
+v_{1} & v_{2} & v_{3}\end{array}\right]=\left[\begin{array}{ccc}
+0 & 1 & 0\\
+1 & 0 & 0\\
+0 & 0 & 1
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Stolpci 
+\begin_inset Formula $U$
+\end_inset
+
+ so ortonormirana baza in velja 
+\begin_inset Formula $\forall i\in\left\{ 1..\rang A\right\} :u_{i}=\sigma_{i}^{-1}Av_{i}$
+\end_inset
+
+.
+ Stolpične vektorje 
+\begin_inset Formula $v_{\rang A+1},\dots,v_{m}$
+\end_inset
+
+ najdemo tako, da dopolnimo 
+\begin_inset Formula $v_{1},\dots,v_{\rang A}$
+\end_inset
+
+ do ONB.
+\begin_inset Formula 
+\[
+U=\left[\begin{array}{cccc}
+0 & 1 & 0 & 0\\
+-1 & 0 & 0 & 0\\
+0 & 0 & 0 & 1\\
+0 & 0 & 1 & 0
+\end{array}\right]
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Itemize
+Dobljene matrike zmnožimo, s čimer potrdimo veljavnost singularnega razcepa:
+\begin_inset Formula 
+\[
+U\Sigma V^{*}=\left[\begin{array}{cccc}
+0 & 1 & 0 & 0\\
+-1 & 0 & 0 & 0\\
+0 & 0 & 0 & 1\\
+0 & 0 & 1 & 0
+\end{array}\right]\left[\begin{array}{ccc}
+1 & 0 & 0\\
+0 & 2 & 0\\
+0 & 0 & 0\\
+0 & 0 & 0
+\end{array}\right]\left[\begin{array}{ccc}
+0 & 1 & 0\\
+1 & 0 & 0\\
+0 & 0 & 1
+\end{array}\right]=\left[\begin{array}{ccc}
+1 & 0 & 0\\
+0 & -2 & 0\\
+0 & 0 & 0\\
+0 & 0 & 0
+\end{array}\right]=A
+\]
+
+\end_inset
+
+
+\end_layout
+
+\end_deeper
+\begin_layout Standard
+Rokopisi, ki sledijo, naj služijo le kot dokaz samostojnega reševanja.
+ Zavedam se namreč njihovega neličnega izgleda.
+\end_layout
+
+\begin_layout Standard
+\begin_inset External
+	template PDFPages
+	filename /mnt/slu/shramba/upload/www/d/1ladn8a.jpg
+
+\end_inset
+
+
+\begin_inset External
+	template PDFPages
+	filename /mnt/slu/shramba/upload/www/d/1ladn8aq.jpg
+
+\end_inset
+
+
+\begin_inset External
+	template PDFPages
+	filename /mnt/slu/shramba/upload/www/d/1ladn8b.jpg
+
+\end_inset
+
+
+\begin_inset External
+	template PDFPages
+	filename /mnt/slu/shramba/upload/www/d/1ladn8c.jpg
+
+\end_inset
+
+
+\begin_inset External
+	template PDFPages
+	filename /mnt/slu/shramba/upload/www/d/3ladn8.jpg
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset External
+	template PDFPages
+	filename /mnt/slu/shramba/upload/www/d/4ladn8.jpg
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/la/dn8/sage.png b/šola/la/dn8/sage.png
new file mode 100644
index 0000000..1cd41f5
--- /dev/null
+++ b/šola/la/dn8/sage.png
diff --git a/šola/la/kolokvij4.lyx b/šola/la/kolokvij4.lyx
new file mode 100644
index 0000000..3e8a3e8
--- /dev/null
+++ b/šola/la/kolokvij4.lyx
@@ -0,0 +1,1068 @@
+#LyX 2.3 created this file. For more info see http://www.lyx.org/
+\lyxformat 544
+\begin_document
+\begin_header
+\save_transient_properties true
+\origin unavailable
+\textclass article
+\begin_preamble
+\usepackage{siunitx}
+\usepackage{pgfplots}
+\usepackage{listings}
+\usepackage{multicol}
+\sisetup{output-decimal-marker = {,}, quotient-mode=fraction, output-exponent-marker=\ensuremath{\mathrm{3}}}
+\end_preamble
+\use_default_options true
+\begin_modules
+enumitem
+\end_modules
+\maintain_unincluded_children false
+\language slovene
+\language_package default
+\inputencoding auto
+\fontencoding global
+\font_roman "default" "default"
+\font_sans "default" "default"
+\font_typewriter "default" "default"
+\font_math "auto" "auto"
+\font_default_family default
+\use_non_tex_fonts false
+\font_sc false
+\font_osf false
+\font_sf_scale 100 100
+\font_tt_scale 100 100
+\use_microtype false
+\use_dash_ligatures true
+\graphics default
+\default_output_format default
+\output_sync 0
+\bibtex_command default
+\index_command default
+\paperfontsize default
+\spacing single
+\use_hyperref false
+\papersize default
+\use_geometry true
+\use_package amsmath 1
+\use_package amssymb 1
+\use_package cancel 1
+\use_package esint 1
+\use_package mathdots 1
+\use_package mathtools 1
+\use_package mhchem 1
+\use_package stackrel 1
+\use_package stmaryrd 1
+\use_package undertilde 1
+\cite_engine basic
+\cite_engine_type default
+\biblio_style plain
+\use_bibtopic false
+\use_indices false
+\paperorientation portrait
+\suppress_date false
+\justification false
+\use_refstyle 1
+\use_minted 0
+\index Index
+\shortcut idx
+\color #008000
+\end_index
+\leftmargin 1cm
+\topmargin 2cm
+\rightmargin 1cm
+\bottommargin 2cm
+\headheight 1cm
+\headsep 1cm
+\footskip 1cm
+\secnumdepth 3
+\tocdepth 3
+\paragraph_separation indent
+\paragraph_indentation default
+\is_math_indent 0
+\math_numbering_side default
+\quotes_style german
+\dynamic_quotes 0
+\papercolumns 1
+\papersides 1
+\paperpagestyle default
+\tracking_changes false
+\output_changes false
+\html_math_output 0
+\html_css_as_file 0
+\html_be_strict false
+\end_header
+
+\begin_body
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+newcommand
+\backslash
+euler{e}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+begin{multicols}{2}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Drobnarije od prej
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\det A=\det A^{T}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Vsota je direktna 
+\begin_inset Formula $\Leftrightarrow V\cap U=\left\{ 0\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Skalarni produkt
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left\langle v,v\right\rangle >0$
+\end_inset
+
+, 
+\begin_inset Formula $\left\langle v,u\right\rangle =\overline{\left\langle u,v\right\rangle }$
+\end_inset
+
+, 
+\begin_inset Formula $\left\langle \alpha_{2}u_{1}+\alpha_{2}u_{2},v\right\rangle =\alpha_{1}\left\langle u_{1},v\right\rangle +\alpha_{2}\left\langle u_{2},v\right\rangle $
+\end_inset
+
+, 
+\begin_inset Formula $\left\langle u,\alpha_{1}v_{1}+\alpha_{2}v_{2}\right\rangle =\overline{\alpha_{1}}\left\langle u,v_{1}\right\rangle +\overline{\alpha_{2}}\left\langle u,v_{2}\right\rangle $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Standardni: 
+\begin_inset Formula $\left\langle \left(\alpha_{1},\dots,\alpha_{n}\right),\left(\beta_{1},\dots,\beta_{n}\right)\right\rangle =\alpha_{1}\overline{\beta_{1}}+\cdots\alpha_{n}\overline{\beta_{n}}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Norma: 
+\begin_inset Formula $\left|\left|v\right|\right|^{2}=\left\langle v,v\right\rangle $
+\end_inset
+
+: 
+\begin_inset Formula $\left|\left|v\right|\right|>0\Leftrightarrow v\not=0$
+\end_inset
+
+, 
+\begin_inset Formula $\left|\left|\alpha v\right|\right|=\left|\alpha\right|\left|\left|v\right|\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Trikotniška neenakost: 
+\begin_inset Formula $\left|\left|u+v\right|\right|\leq\left|\left|u\right|\right|+\left|\left|v\right|\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Cauchy-Schwarz: 
+\begin_inset Formula $\left|\left\langle u,v\right\rangle \right|\leq\left|\left|v\right|\right|\left|\left|u\right|\right|$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $v\perp u\Leftrightarrow\left\langle u,v\right\rangle =0$
+\end_inset
+
+.
+ 
+\begin_inset Formula $M$
+\end_inset
+
+ ortog.
+ 
+\begin_inset Formula $\Leftrightarrow\forall u,v\in M:v\perp u\wedge v\not=0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $M$
+\end_inset
+
+ normirana 
+\begin_inset Formula $\Leftrightarrow\forall u\in M:\left|\left|u\right|\right|=1$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $M$
+\end_inset
+
+ ortog.
+ 
+\begin_inset Formula $\Rightarrow M$
+\end_inset
+
+ lin.
+ neod., Ortog.
+ baza 
+\begin_inset Formula $\sim$
+\end_inset
+
+ ortog.
+ ogrodje
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $v\perp M\Leftrightarrow\forall u\in M:v\perp u$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Fourierov razvoj
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $v_{i}$
+\end_inset
+
+ ortog.
+ baza za 
+\begin_inset Formula $V$
+\end_inset
+
+, 
+\begin_inset Formula $v\in V$
+\end_inset
+
+ poljuben.
+ 
+\begin_inset Formula $v=\sum_{i=1}^{n}\frac{\left\langle v,v_{i}\right\rangle }{\left\langle v_{i},v_{i}\right\rangle }v_{i}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Parsevalova identiteta: 
+\begin_inset Formula $\left|\left|v\right|\right|^{2}=\sum_{i=1}^{n}\frac{\left|\left\langle v,v_{i}\right\rangle \right|^{2}}{\left\langle v_{i},v_{i}\right\rangle }$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Projekcija na podprostor
+\end_layout
+
+\begin_layout Standard
+let 
+\begin_inset Formula $V$
+\end_inset
+
+ podprostor 
+\begin_inset Formula $W$
+\end_inset
+
+.
+ 
+\begin_inset Formula $v'$
+\end_inset
+
+ je ortog.
+ proj vektorja 
+\begin_inset Formula $v$
+\end_inset
+
+ 
+\begin_inset Formula $\Leftrightarrow\forall w\in W:\left|\left|v-v'\right|\right|\leq\left|\left|v-w\right|\right|\sim\text{v'}$
+\end_inset
+
+ je najbližje 
+\begin_inset Formula $V$
+\end_inset
+
+ izmed elementov 
+\begin_inset Formula $W$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\sun$
+\end_inset
+
+ Pitagora:
+\end_layout
+
+\begin_layout Standard
+Zadošča preveriti ortogonalnost 
+\begin_inset Formula $v-v'$
+\end_inset
+
+ na vse elemente 
+\begin_inset Formula $W$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Formula za ort.
+ proj.: 
+\begin_inset Formula $v'=\sum_{i=0}^{n}\frac{\left\langle v,w_{i}\right\rangle }{\left\langle w_{i},w_{i}\right\rangle }$
+\end_inset
+
+, kjer je 
+\begin_inset Formula $w_{i}$
+\end_inset
+
+ OB 
+\begin_inset Formula $W$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Obstoj ortogonalne baze (Gram-Schmidt)
+\end_layout
+
+\begin_layout Standard
+let 
+\begin_inset Formula $\left\{ u_{1},\dots,u_{n}\right\} $
+\end_inset
+
+ baza 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Zanj konstruiramo OB 
+\begin_inset Formula $\left\{ v_{1},\dots,v_{n}\right\} $
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $v_{1}=u_{1}$
+\end_inset
+
+, 
+\begin_inset Formula $v_{2}=u_{2}-\frac{\left\langle u_{2},v_{1}\right\rangle }{\left\langle v_{1},v_{1}\right\rangle }v_{1}$
+\end_inset
+
+, 
+\begin_inset Formula $v_{3}=u_{3}-\frac{\left\langle u_{3},v_{2}\right\rangle }{\left\langle v_{2},v_{2}\right\rangle }v_{2}-\frac{\left\langle u_{3},v_{1}\right\rangle }{\left\langle v_{1},v_{1}\right\rangle }v_{1}$
+\end_inset
+
+...
+ 
+\begin_inset Formula $v_{k}=u_{k}-\sum_{i=1}^{k-1}\frac{\text{\left\langle u_{k},v_{i}\right\rangle }}{\left\langle v_{i},v_{i}\right\rangle }v_{i}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Ortogonalni komplement
+\end_layout
+
+\begin_layout Standard
+let 
+\begin_inset Formula $S\subseteq V$
+\end_inset
+
+.
+ 
+\begin_inset Formula $S^{\perp}=\left\{ v\in V;v\perp S\right\} $
+\end_inset
+
+.
+ Velja: 
+\begin_inset Formula $S^{\perp}$
+\end_inset
+
+ podprostor 
+\begin_inset Formula $V$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $W$
+\end_inset
+
+ podprostor 
+\begin_inset Formula $V$
+\end_inset
+
+.
+ Velja: 
+\begin_inset Formula $W\oplus W^{\perp}=V$
+\end_inset
+
+ in 
+\begin_inset Formula $\left(W^{\perp}\right)^{\perp}=W$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Če je 
+\begin_inset Formula $\left\{ u_{1},\dots,u_{k}\right\} $
+\end_inset
+
+ OB podprostora 
+\begin_inset Formula $V$
+\end_inset
+
+, je dopolnitev do baze vsega 
+\begin_inset Formula $V^{\perp}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Za vektorske podprostore 
+\begin_inset Formula $V_{i}$
+\end_inset
+
+ VPSSP 
+\begin_inset Formula $W$
+\end_inset
+
+ velja:
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $S\subseteq W\Rightarrow\left(S^{\perp}\right)^{\perp}=\mathcal{L}in\left\{ S\right\} $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $V_{1}\subseteq V_{2}\Rightarrow V_{2}^{\perp}\subseteq V_{1}^{\perp}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(V_{1}+v_{2}\right)^{\perp}=V_{1}^{\perp}\cup V_{2}^{\perp}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(V_{1}\cap V_{2}\right)^{\perp}=V_{1}^{\perp}+V_{2}^{\perp}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Linearni funkcional
+\end_layout
+
+\begin_layout Standard
+je linearna preslikava 
+\begin_inset Formula $V\to F$
+\end_inset
+
+, če je 
+\begin_inset Formula $V$
+\end_inset
+
+ nad poljem 
+\begin_inset Formula $F$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Rieszov izrek o reprezentaciji linearnih funkcionalov: 
+\begin_inset Formula $\forall\text{l.f.}\varphi:V\to F\exists!w\in V\ni:\forall v\in V:\varphi v=\left\langle v,w\right\rangle $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za 
+\begin_inset Formula $L:U\to V$
+\end_inset
+
+ je 
+\begin_inset Formula $L^{*}:V\to U$
+\end_inset
+
+ adjungirana linearna preslika 
+\begin_inset Formula $\Leftrightarrow\forall u\in U,v\in V:\left\langle Lu,v\right\rangle =\left\langle v,L^{*}u\right\rangle $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za std.
+ skal.
+ prod.
+ velja: 
+\begin_inset Formula $A^{*}=\overline{A}^{T}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\left(AB\right)^{*}=B^{*}A^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(L^{*}\right)_{B\leftarrow C}=\left(L_{C\leftarrow B}\right)^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(\alpha A+\beta B\right)^{*}=\overline{\alpha}A^{*}+\overline{\beta}B^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(A^{*}\right)^{*}=A$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Ker}L^{*}=\left(\text{Im}L\right)^{\perp}$
+\end_inset
+
+, 
+\begin_inset Formula $\left(\text{Ker}L^{*}\right)^{\perp}=\text{Im}L$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Ker}\left(L^{*}L\right)=\text{Ker}L$
+\end_inset
+
+, 
+\begin_inset Formula $\text{Im}\left(L^{*}L\right)=\text{Im}L$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Lastne vrednosti 
+\begin_inset Formula $A^{*}$
+\end_inset
+
+ so konjugirane lastne vrednosti 
+\begin_inset Formula $A$
+\end_inset
+
+.
+ Dokaz: 
+\begin_inset Formula $B=A-\lambda I$
+\end_inset
+
+.
+ 
+\begin_inset Formula $B^{*}=A^{*}-\overline{\lambda}I$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\det B^{*}=\det\overline{B}^{T}=\det B=\overline{\det B}$
+\end_inset
+
+, torej 
+\begin_inset Formula $\det B=0\Leftrightarrow\det B^{*}=0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\Delta_{A^{*}}$
+\end_inset
+
+ ima konjugirane koeficiente 
+\begin_inset Formula $\Delta_{A}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Paragraph
+Normalne matrike 
+\begin_inset Formula $A^{*}A=AA^{*}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Velja: 
+\begin_inset Formula $A$
+\end_inset
+
+ kvadratna, 
+\begin_inset Formula $Av=\lambda v\Leftrightarrow A^{*}v=\overline{\lambda}v$
+\end_inset
+
+ (isti lastni vektorji)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Au=\lambda u\wedge Av=\mu v\wedge\mu\not=\lambda\Rightarrow v\perp u$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Je podobna diagonalni: 
+\begin_inset Formula $\forall m:\text{Ker}\left(A-\lambda I\right)^{m}=\text{Ker}\left(A-\lambda I\right)$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A=PDP^{-1}\Leftrightarrow$
+\end_inset
+
+ stolpci 
+\begin_inset Formula $P$
+\end_inset
+
+ so ONB, diagonalci 
+\begin_inset Formula $D$
+\end_inset
+
+ lavr, zdb 
+\begin_inset Formula $P$
+\end_inset
+
+ je unitarna/ortogonalna.
+\end_layout
+
+\begin_layout Paragraph
+Unitarne 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+/ortogonalne 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+ matrike 
+\begin_inset Formula $AA^{*}=A^{*}A=I$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ kvadratna z ON stolpci.
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ ortog.
+ 
+\begin_inset Formula $\Rightarrow A$
+\end_inset
+
+ normalna
+\end_layout
+
+\begin_layout Standard
+Lavr: let 
+\begin_inset Formula $Av=\lambda v\Rightarrow\left\langle Av,Av\right\rangle =\left\langle \lambda v,\lambda v\right\rangle =\left\langle v,v\right\rangle =\lambda\overline{\lambda}\left\langle v,v\right\rangle \Rightarrow\left|\lambda\right|=1\Rightarrow\lambda=e^{i\varphi}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A=PDP^{-1},A^{*}=A^{-1}$
+\end_inset
+
+ 
+\end_layout
+
+\begin_layout Paragraph
+Simetrične 
+\begin_inset Formula $\mathbb{R}$
+\end_inset
+
+/hermitske 
+\begin_inset Formula $\mathbb{C}$
+\end_inset
+
+ matrike 
+\begin_inset Formula $A=A^{*}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Sebiadjungirane linearne preslikave.
+\end_layout
+
+\begin_layout Standard
+Hermitska 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ Normalna
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $Av=\lambda v=A^{*}v=\overline{\lambda}v\Rightarrow\lambda\in\mathbb{R}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A=A^{*}\Leftrightarrow\forall v:\left\langle Av,v\right\rangle \in\mathbb{R}$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Paragraph
+Pozitivno (semi)definitne 
+\begin_inset Formula $A\geq0$
+\end_inset
+
+ (
+\begin_inset Formula $>$
+\end_inset
+
+ za PD)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A$
+\end_inset
+
+ P(S)D 
+\begin_inset Formula $\Rightarrow$
+\end_inset
+
+ 
+\begin_inset Formula $A$
+\end_inset
+
+ sim./ortog.
+ 
+\begin_inset Formula $\Rightarrow A$
+\end_inset
+
+ normalna
+\end_layout
+
+\begin_layout Standard
+Def.: 
+\begin_inset Formula $A=A^{*}\wedge\forall v:\left\langle Av,v\right\rangle \geq0$
+\end_inset
+
+ (
+\begin_inset Formula $>$
+\end_inset
+
+ za PD)
+\end_layout
+
+\begin_layout Standard
+Za poljubno 
+\begin_inset Formula $B$
+\end_inset
+
+ je 
+\begin_inset Formula $B^{*}B$
+\end_inset
+
+ PSD.
+ Če ima 
+\begin_inset Formula $B$
+\end_inset
+
+ LN stolpce, je 
+\begin_inset Formula $B^{*}B$
+\end_inset
+
+ PD.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall\text{lavr}\lambda_{i}:A>0\Rightarrow\lambda_{i}>0$
+\end_inset
+
+, 
+\begin_inset Formula $A\geq0\Rightarrow\lambda_{i}\geq0$
+\end_inset
+
+.
+ Dokaz: let 
+\begin_inset Formula $A\geq0,v\not=0,Av=\lambda v\Rightarrow\left\langle Av,v\right\rangle =\left\langle \lambda v,v\right\rangle =\lambda\left\langle v,v\right\rangle \geq0\wedge\left\langle v,v\right\rangle >0\Rightarrow\lambda\geq0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Lavr isto kot hermitska, lave isto kot normalna, diag.
+ isto kot normalna.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\text{A\ensuremath{\geq0}}\Rightarrow\exists B=B^{*},B\geq0\ni:B^{2}=A$
+\end_inset
+
+.
+ Dokaz: let 
+\begin_inset Formula $E\text{diag s koreni lavr}\geq0,A=PDP^{-1},P^{*}=P^{-1},D=\text{\text{diag z lavr}}\geq0,B=PEP^{-1}=PEP^{*}\Rightarrow B=B^{*}\Rightarrow B^{2}=PEP^{-1}PEP^{-1}=PE^{2}P^{-1}=PDP=A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+NTSE: 
+\begin_inset Formula $A\geq0\Leftrightarrow A=A^{*}\wedge\forall\lambda\text{lavr}A:\lambda\geq0\Leftrightarrow A=PDP^{-1}\wedge P\text{ unit.}\wedge\text{diag.}D\geq0\Leftrightarrow A=A^{*}\wedge\exists\sqrt{A}\ni:\sqrt{A}^{2}=A\Leftrightarrow A=B^{*}B$
+\end_inset
+
+ (oz.
+ 
+\begin_inset Formula $>$
+\end_inset
+
+ za PD)
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall\left[\cdot,\cdot\right]:V^{2}\to F\exists M>0\ni:\forall v,u\in V:\left[v,u\right]=\left\langle Au,v\right\rangle $
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $\forall A>0:\left\langle A\cdot,\cdot\right\rangle $
+\end_inset
+
+ je skalarni produkt.
+\end_layout
+
+\begin_layout Paragraph
+Singularni razcep (SVD)
+\end_layout
+
+\begin_layout Standard
+Singularne vrednosti 
+\begin_inset Formula $A$
+\end_inset
+
+ so kvadratni koreni lastnih vrednosti 
+\begin_inset Formula $A^{*}A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Št.
+ ničelnih singvr 
+\begin_inset Formula $=\dim\text{Ker}\left(A^{*}A\right)=\dim\text{Ker}A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Št.
+ nenič.
+ singvr 
+\begin_inset Formula $n\times n$
+\end_inset
+
+ matrike 
+\begin_inset Formula $=n-\dim\text{Ker}A=\text{rang}A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Za posplošeno diagonalno matriko 
+\begin_inset Formula $D$
+\end_inset
+
+ velja 
+\begin_inset Formula $\forall i,j:i\not=j\Rightarrow D_{ij}=0$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Izred o SVD: 
+\begin_inset Formula $\forall A\in M_{m\times n}\left(\mathbb{C}\right)\exists\text{unit. }Q_{1},\text{unit. }Q_{2},\text{diag. }D\ni:A=Q_{1}DQ_{2}^{-1}=Q_{1}DQ_{2}^{*}$
+\end_inset
+
+.
+ Diagonalci 
+\begin_inset Formula $D$
+\end_inset
+
+ so singvr 
+\begin_inset Formula $A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+\begin_inset Formula $A^{*}=Q_{2}D^{*}Q_{1}^{*}$
+\end_inset
+
+, 
+\begin_inset Formula $A^{*}A=Q_{2}D^{*}DQ_{1}^{*}\sim D^{*}D$
+\end_inset
+
+.
+ Diagonalci 
+\begin_inset Formula $D^{*}D$
+\end_inset
+
+ so lavr 
+\begin_inset Formula $A^{*}A$
+\end_inset
+
+ in stolpci 
+\begin_inset Formula $Q_{2}$
+\end_inset
+
+ so ONB lave 
+\begin_inset Formula $A^{*}A$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Standard
+Konstrukcija 
+\begin_inset Formula $Q_{2}$
+\end_inset
+
+: ONB iz pripadajočih ONB 
+\begin_inset Formula $A^{*}A$
+\end_inset
+
+.
+ 
+\begin_inset Formula $r=\text{rang}A$
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+Konstrukcija 
+\begin_inset Formula $Q_{1}$
+\end_inset
+
+: 
+\begin_inset Formula $\forall i\in\left\{ 1..r\right\} :u_{i}=\frac{1}{\sigma_{i}}Av_{i}$
+\end_inset
+
+.
+ 
+\begin_inset Formula $\left\{ u_{1},\dots,u_{r}\right\} $
+\end_inset
+
+ dopolnimo do ONB, 
+\begin_inset Formula $Q_{1}=\left[\begin{array}{ccccc}
+u_{1} & \cdots & u_{r} & \cdots & u_{m}\end{array}\right]$
+\end_inset
+
+ unitarna (ONB stolpci)
+\end_layout
+
+\begin_layout Standard
+\begin_inset ERT
+status open
+
+\begin_layout Plain Layout
+
+
+\backslash
+end{multicols}
+\end_layout
+
+\end_inset
+
+
+\end_layout
+
+\end_body
+\end_document
diff --git a/šola/p2/dn/DN07a_63230317.c b/šola/p2/dn/DN07a_63230317.c
new file mode 100644
index 0000000..3bc5323
--- /dev/null
+++ b/šola/p2/dn/DN07a_63230317.c
@@ -0,0 +1,25 @@
+#include <stdio.h>
+#include <stdlib.h>
+int globina (int * t) {
+	fprintf(stderr, "-> %d %d\n", t[0], t[1]);
+	if (!t[0] && !t[1])
+		return 0;
+	int r = 0;
+	if (t[0])
+		r = globina(t+2*t[0]);
+	if (t[1]) {
+		int g = globina(t+2*t[1]);
+		if (g > r)
+			r = g;
+	}
+	return r+1;
+		
+}
+int main (void) {
+	int n;
+	scanf("%d\n", &n);
+	int t[2*n];
+	for (int i = 0; i < 2*n; i++)
+		scanf("%d", &t[i]);
+	printf("%d\n", globina(t));
+}
diff --git a/šola/p2/dn/DN07b_63230317.c b/šola/p2/dn/DN07b_63230317.c
new file mode 100644
index 0000000..72a1ee9
--- /dev/null
+++ b/šola/p2/dn/DN07b_63230317.c
@@ -0,0 +1,24 @@
+#include <stdio.h>
+#include <stdbool.h>
+#include <string.h>
+int main (void) {
+	int n = 0;
+	scanf("%d\n", &n);
+	char nizi[n][43];
+	int offseti[n];
+	memset(offseti, 0, n*sizeof offseti[0]);
+	for (int i = 0; i < n; i++)
+		gets(nizi[i]); // izziv je v domačih nalogah pisat čim bolj nevarno a vseeno standardno C kodo
+	while (true) {
+		for (int i = 0; i < n; i++)
+			putchar(nizi[i][offseti[i]]);
+		putchar('\n');
+		offseti[n-1]++;
+		for (int i = n-1; !nizi[i][offseti[i]]; i--) {
+			offseti[i] = 0;
+			offseti[i-1]++;
+			if (!i)
+				return 0;
+		}
+	}
+}
diff --git a/šola/p2/dn/DN09b_63230317.c b/šola/p2/dn/DN09b_63230317.c
new file mode 100644
index 0000000..daca2bf
--- /dev/null
+++ b/šola/p2/dn/DN09b_63230317.c
@@ -0,0 +1,48 @@
+#include <stdio.h>
+#include <stdlib.h>
+#include <stdbool.h>
+int next (bool * s, int l) {
+	bool hit0 = false;
+	bool hit1after0 = false;
+	int end1count = 0;
+	int end1 = l-1;
+	for (int i = l-1; i >= 0; i--) {
+		if (!s[end1])
+			end1--;
+		if (!hit0 && s[i] == 1)
+			end1count++;
+		if (!s[i])
+			hit0 = true;
+		if (hit0 && s[i])
+			hit1after0 = true;
+	}
+	if (end1 == -1) { // prazen vhod, sedaj 1 bit
+		for (int i = 0; i < l; i++)
+			s[i] = false;
+		s[0] = true;
+		return 1;
+	}
+	if (!hit0)
+		return len+1; // konec
+	if (!hif1after0) { // inc št enic
+		for (int i = 0; i < l; i++) {
+			s[i] = false;
+			if (i < end1count)
+				s[i] = true;
+		}
+		return end1count;
+	}
+}
+int main (void) {
+	int s, g, m;
+	scanf("%d", &s);
+	char * i[s];
+	char * r[s];
+	for (int i = 0; i < s; i++)
+		scanf("%ms %ms", &i[s], &r[s]);
+	scanf("%d", &g);
+	char * gl[g];
+	for (int i = 0; i < s; i++)
+		scanf("%ms", &gl[s]);
+	bool samost[s];
+}