diff options
Diffstat (limited to '')
| -rw-r--r-- | vorlesungen/11_msegraphen/Makefile | 33 | ||||
| -rw-r--r-- | vorlesungen/11_msegraphen/MathSemMSE-11-graphen.tex | 14 | ||||
| -rw-r--r-- | vorlesungen/11_msegraphen/common.tex | 16 | ||||
| -rw-r--r-- | vorlesungen/11_msegraphen/graphen-handout.tex | 11 | ||||
| -rw-r--r-- | vorlesungen/11_msegraphen/slides.tex | 31 | ||||
| -rw-r--r-- | vorlesungen/slides/8/Makefile.inc | 3 | ||||
| -rw-r--r-- | vorlesungen/slides/8/chapter.tex | 4 | ||||
| -rw-r--r-- | vorlesungen/slides/8/chrind.tex | 231 | ||||
| -rw-r--r-- | vorlesungen/slides/8/chrindprop.tex | 62 | ||||
| -rw-r--r-- | vorlesungen/slides/8/chroma1.tex | 56 | ||||
| -rw-r--r-- | vorlesungen/slides/8/spanningtree.tex | 4 |
11 files changed, 464 insertions, 1 deletions
diff --git a/vorlesungen/11_msegraphen/Makefile b/vorlesungen/11_msegraphen/Makefile new file mode 100644 index 0000000..36f1877 --- /dev/null +++ b/vorlesungen/11_msegraphen/Makefile @@ -0,0 +1,33 @@ +# +# Makefile -- graphen +# +# (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +# +all: graphen-handout.pdf MathSemMSE-11-graphen.pdf + +include ../slides/Makefile.inc + +SOURCES = common.tex slides.tex $(slides) + +MathSemMSE-11-graphen.pdf: MathSemMSE-11-graphen.tex $(SOURCES) + pdflatex MathSemMSE-11-graphen.tex + +graphen-handout.pdf: graphen-handout.tex $(SOURCES) + pdflatex graphen-handout.tex + +thumbnail: thumbnail.jpg # fix1.jpg + +thumbnail.pdf: MathSemMSE-11-graphen.pdf + pdfjam --outfile thumbnail.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-11-graphen.pdf 1 +thumbnail.jpg: thumbnail.pdf + convert -density 300 thumbnail.pdf \ + -resize 1920x1080 -units PixelsPerInch thumbnail.jpg + +fix1.pdf: MathSemMSE-11-graphen.pdf + pdfjam --outfile fix1.pdf --papersize '{16cm,9cm}' \ + MathSemMSE-11-graphen.pdf 1 +fix1.jpg: fix1.pdf + convert -density 300 fix1.pdf \ + -resize 1920x1080 -units PixelsPerInch fix1.jpg + diff --git a/vorlesungen/11_msegraphen/MathSemMSE-11-graphen.tex b/vorlesungen/11_msegraphen/MathSemMSE-11-graphen.tex new file mode 100644 index 0000000..6d4ff2d --- /dev/null +++ b/vorlesungen/11_msegraphen/MathSemMSE-11-graphen.tex @@ -0,0 +1,14 @@ +% +% MathSem-11-msegraphen.tex -- Präsentation +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{true} +\begin{document} +\begin{frame} +\titlepage +\end{frame} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/11_msegraphen/common.tex b/vorlesungen/11_msegraphen/common.tex new file mode 100644 index 0000000..67cf5e5 --- /dev/null +++ b/vorlesungen/11_msegraphen/common.tex @@ -0,0 +1,16 @@ +% +% common.tex -- gemeinsame definition +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\input{../common/packages.tex} +\input{../common/common.tex} +\mode<beamer>{% +\usetheme[hideothersubsections,hidetitle]{Hannover} +} +\beamertemplatenavigationsymbolsempty +\title[Graphen]{Spektrale Graphentheorie} +\author[A.~Müller]{Prof. Dr. Andreas Müller} +\date[]{} +\newboolean{presentation} + diff --git a/vorlesungen/11_msegraphen/graphen-handout.tex b/vorlesungen/11_msegraphen/graphen-handout.tex new file mode 100644 index 0000000..58d0aff --- /dev/null +++ b/vorlesungen/11_msegraphen/graphen-handout.tex @@ -0,0 +1,11 @@ +% +% msegraphen-handout.tex -- Handout XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% +\documentclass[handout,aspectratio=169]{beamer} +\input{common.tex} +\setboolean{presentation}{false} +\begin{document} +\input{slides.tex} +\end{document} diff --git a/vorlesungen/11_msegraphen/slides.tex b/vorlesungen/11_msegraphen/slides.tex new file mode 100644 index 0000000..b152d82 --- /dev/null +++ b/vorlesungen/11_msegraphen/slides.tex @@ -0,0 +1,31 @@ +% +% slides.tex -- XXX +% +% (c) 2017 Prof Dr Andreas Müller, Hochschule Rapperswil +% + +%\folie{8/graph.tex} +%\folie{8/dgraph.tex} +%\folie{8/grad.tex} +%\folie{8/inzidenz.tex} +%\folie{8/inzidenzd.tex} +%\folie{8/diffusion.tex} +%\folie{8/laplace.tex} +%\folie{8/produkt.tex} +%\folie{8/fourier.tex} +%\folie{8/spanningtree.tex} + +%\folie{8/pfade/adjazenz.tex} +%\folie{8/pfade/langepfade.tex} +%\folie{8/pfade/beispiel.tex} +%\folie{8/pfade/gf.tex} + +%\folie{8/floyd-warshall/problem.tex} +%\folie{8/floyd-warshall/rekursion.tex} +%\folie{8/floyd-warshall/iteration.tex} +%\folie{8/floyd-warshall/wegiteration.tex} +%\folie{8/floyd-warshall/wege.tex} + +%\folie{8/chrind.tex} +%\folie{8/chrindprop.tex} +\folie{8/chroma1.tex} diff --git a/vorlesungen/slides/8/Makefile.inc b/vorlesungen/slides/8/Makefile.inc index d46dc7f..b2823ca 100644 --- a/vorlesungen/slides/8/Makefile.inc +++ b/vorlesungen/slides/8/Makefile.inc @@ -28,5 +28,8 @@ chapter8 = \ ../slides/8/tokyo/bahn0.tex \ ../slides/8/tokyo/bahn1.tex \ ../slides/8/tokyo/bahn2.tex \ + ../slides/8/chrind.tex \ + ../slides/8/chrindprop.tex \ + ../slides/8/chroma1.tex \ ../slides/8/chapter.tex diff --git a/vorlesungen/slides/8/chapter.tex b/vorlesungen/slides/8/chapter.tex index 6a0b13f..2110bb2 100644 --- a/vorlesungen/slides/8/chapter.tex +++ b/vorlesungen/slides/8/chapter.tex @@ -30,3 +30,7 @@ \folie{8/tokyo/bahn1.tex} \folie{8/tokyo/bahn2.tex} +\folie{8/chrind.tex} +\folie{8/chrindprop.tex} +\folie{8/chroma1.tex} + diff --git a/vorlesungen/slides/8/chrind.tex b/vorlesungen/slides/8/chrind.tex new file mode 100644 index 0000000..bd406ab --- /dev/null +++ b/vorlesungen/slides/8/chrind.tex @@ -0,0 +1,231 @@ +% +% chrind.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Chromatische Zahl und Unabhängigkeitszahl} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.48\textwidth} +\begin{block}{Chromatische Zahl} +$\operatorname{chr}(G)=\mathstrut$ +minimale Anzahl Farben, die zum Einfärben eines Graphen $G$ nötig sind derart, +dass benachbarte Knoten verschiedene Farben haben. +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\Ra{2} +\def\Ri{1} +\def\e{1.0} +\def\r{0.2} + +\definecolor{rot}{rgb}{0.8,0,0.8} +\definecolor{gruen}{rgb}{0.2,0.6,0.2} +\definecolor{blau}{rgb}{1,0.6,0.2} + +\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)}); +\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)}); +\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)}); +\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)}); +\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)}); + +\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)}); +\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)}); +\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)}); +\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)}); +\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)}); + +\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle; +\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle; +\draw (PA)--(QA); +\draw (PB)--(QB); +\draw (PC)--(QC); +\draw (PD)--(QD); +\draw (PE)--(QE); + +\only<1>{ + \fill[color=white] (PA) circle[radius=\r]; + \fill[color=white] (PB) circle[radius=\r]; + \fill[color=white] (PC) circle[radius=\r]; + \fill[color=white] (PD) circle[radius=\r]; + \fill[color=white] (PE) circle[radius=\r]; + \fill[color=white] (QA) circle[radius=\r]; + \fill[color=white] (QB) circle[radius=\r]; + \fill[color=white] (QC) circle[radius=\r]; + \fill[color=white] (QD) circle[radius=\r]; + \fill[color=white] (QE) circle[radius=\r]; +} + +\only<2->{ + \fill[color=blau] (PA) circle[radius=\r]; + \fill[color=rot] (PB) circle[radius=\r]; + \fill[color=rot] (PC) circle[radius=\r]; + \fill[color=gruen] (PD) circle[radius=\r]; + \fill[color=gruen] (PE) circle[radius=\r]; + + \fill[color=rot] (QA) circle[radius=\r]; + \fill[color=blau] (QB) circle[radius=\r]; + \fill[color=gruen] (QC) circle[radius=\r]; + \fill[color=rot] (QD) circle[radius=\r]; + \fill[color=blau] (QE) circle[radius=\r]; +} + +\draw (PA) circle[radius=\r]; +\draw (PB) circle[radius=\r]; +\draw (PC) circle[radius=\r]; +\draw (PD) circle[radius=\r]; +\draw (PE) circle[radius=\r]; + +\draw (QA) circle[radius=\r]; +\draw (QB) circle[radius=\r]; +\draw (QC) circle[radius=\r]; +\draw (QD) circle[radius=\r]; +\draw (QE) circle[radius=\r]; + +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{chr} G = 3$}; + +\end{tikzpicture} +\end{center} +\end{block} +\end{column} +\begin{column}{0.48\textwidth} +\uncover<3->{% +\begin{block}{Unabhängigkeitszahl} +$\operatorname{ind}(G)=\mathstrut$ +maximale Anzahl nicht benachbarter Knoten +\begin{center} +\begin{tikzpicture}[>=latex,thick] + +\def\Ra{2} +\def\Ri{1} +\def\e{1.0} +\def\r{0.2} + +\definecolor{rot}{rgb}{0.8,0,0.8} +\definecolor{gruen}{rgb}{0.2,0.6,0.2} +\definecolor{blau}{rgb}{1,0.6,0.2} +\definecolor{gelb}{rgb}{0,0,1} + +\coordinate (PA) at ({\Ri*sin(0*72)},{\e*\Ri*cos(0*72)}); +\coordinate (PB) at ({\Ri*sin(1*72)},{\e*\Ri*cos(1*72)}); +\coordinate (PC) at ({\Ri*sin(2*72)},{\e*\Ri*cos(2*72)}); +\coordinate (PD) at ({\Ri*sin(3*72)},{\e*\Ri*cos(3*72)}); +\coordinate (PE) at ({\Ri*sin(4*72)},{\e*\Ri*cos(4*72)}); + +\coordinate (QA) at ({\Ra*sin(0*72)},{\e*\Ra*cos(0*72)}); +\coordinate (QB) at ({\Ra*sin(1*72)},{\e*\Ra*cos(1*72)}); +\coordinate (QC) at ({\Ra*sin(2*72)},{\e*\Ra*cos(2*72)}); +\coordinate (QD) at ({\Ra*sin(3*72)},{\e*\Ra*cos(3*72)}); +\coordinate (QE) at ({\Ra*sin(4*72)},{\e*\Ra*cos(4*72)}); + +\draw (PA)--(PC)--(PE)--(PB)--(PD)--cycle; +\draw (QA)--(QB)--(QC)--(QD)--(QE)--cycle; +\draw (PA)--(QA); +\draw (PB)--(QB); +\draw (PC)--(QC); +\draw (PD)--(QD); +\draw (PE)--(QE); + +\foreach \n in {1,...,7}{ + \only<\n>{\node[color=white] at (1,2.9) {$\n$};} +} + +\fill[color=white] (PA) circle[radius=\r]; +\fill[color=white] (PB) circle[radius=\r]; +\fill[color=white] (PC) circle[radius=\r]; +\fill[color=white] (PD) circle[radius=\r]; +\fill[color=white] (PE) circle[radius=\r]; +\fill[color=white] (QA) circle[radius=\r]; +\fill[color=white] (QB) circle[radius=\r]; +\fill[color=white] (QC) circle[radius=\r]; +\fill[color=white] (QD) circle[radius=\r]; +\fill[color=white] (QE) circle[radius=\r]; + +\only<4->{ + \fill[color=rot] (QA) circle[radius={1.5*\r}]; + \fill[color=rot!40] (QB) circle[radius=\r]; + \fill[color=rot!40] (QE) circle[radius=\r]; + \fill[color=rot!40] (PA) circle[radius=\r]; +} + +\only<5->{ + \fill[color=blau] (PB) circle[radius={1.5*\r}]; + \fill[color=blau!40] (PD) circle[radius=\r]; + \fill[color=blau!40] (PE) circle[radius=\r]; + \fill[color=blau!80,opacity=0.5] (QB) circle[radius=\r]; +} + +\only<6->{ + \fill[color=gruen] (PC) circle[radius={1.5*\r}]; + \fill[color=gruen!40] (QC) circle[radius=\r]; + \fill[color=gruen!80,opacity=0.5] (PA) circle[radius=\r]; + \fill[color=gruen!80,opacity=0.5] (PE) circle[radius=\r]; +} + +\only<7->{ + \fill[color=gelb] (QD) circle[radius={1.5*\r}]; + \fill[color=gelb!80,opacity=0.5] (QC) circle[radius=\r]; + \fill[color=gelb!80,opacity=0.5] (QE) circle[radius=\r]; + \fill[color=gelb!80,opacity=0.5] (PD) circle[radius=\r]; +} + +\only<-3|handout:0>{ + \draw (QA) circle[radius=\r]; +} +\only<4->{ + \draw (QA) circle[radius={1.5*\r}]; +} + +\only<-4|handout:0>{ + \draw (PB) circle[radius=\r]; +} +\only<5->{ + \draw (PB) circle[radius={1.5*\r}]; +} + +\only<-5|handout:0>{ + \draw (PC) circle[radius=\r]; +} +\only<6->{ + \draw (PC) circle[radius={1.5*\r}]; +} + +\only<-6|handout:0>{ + \draw (QD) circle[radius=\r]; +} +\only<7->{ + \draw (QD) circle[radius={1.5*\r}]; +} + +\draw (PA) circle[radius=\r]; +\draw (PD) circle[radius=\r]; +\draw (PE) circle[radius=\r]; + +\draw (QB) circle[radius=\r]; +\draw (QC) circle[radius=\r]; +\draw (QE) circle[radius=\r]; + +\only<4|handout:0>{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 1$}; +} +\only<5|handout:0>{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 2$}; +} +\only<6|handout:0>{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 3$}; +} +\only<7->{ +\node at ($0.5*(QC)+0.5*(QD)+(0,-0.2)$) [below] {$\operatorname{ind} G = 4$}; +} + +\end{tikzpicture} +\end{center} +\end{block}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/chrindprop.tex b/vorlesungen/slides/8/chrindprop.tex new file mode 100644 index 0000000..094588c --- /dev/null +++ b/vorlesungen/slides/8/chrindprop.tex @@ -0,0 +1,62 @@ +% +% chrindprop.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Zusammenhang zwischen $\operatorname{chr}G$ und $\operatorname{ind}G$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.38\textwidth} +\begin{block}{Proposition} +Ist $G$ ein Graph mit $n$ Knoten, dann gilt +\[ +\operatorname{chr}G +\cdot +\operatorname{ind}G +\ge n +\] +\end{block} +\uncover<2->{% +\begin{block}{Beispiel} +Peterson-Graph $K$ hat $n=10$ Knoten: +\[ +\operatorname{chr}(K) +\cdot +\operatorname{ind}(K) += +3\cdot 4 +\ge +10 += +n +\] +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<3->{% +\begin{proof}[Beweis] +\begin{itemize} +\item<4-> eine minimale Färbung hat $\operatorname{chr}(G)$ Farben +\item<5-> Sie teilt die Knoten in $\operatorname{chr}(G)$ +gleichfarbige Mengen auf +\item<6-> Jede einfarbige Menge von Knoten ist unabhängig, d.~h.~sie +besteht aus Knoten, die nicht miteinander verbunden sind. +\item<7-> Jede einfarbige Menge enthält höchstens $\operatorname{ind}(G)$ +\item<8-> Die Gesamtzahl der Knoten ist +\[ +n\uncover<9->{=\sum_{\text{Farbe}}\underbrace{|V_{\text{Farbe}}|}_{\le \operatorname{ind}(G)}} +\uncover<10->{\le +\operatorname{chr}(G) +\cdot +\operatorname{ind}(G)} +\] +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/chroma1.tex b/vorlesungen/slides/8/chroma1.tex new file mode 100644 index 0000000..6a55704 --- /dev/null +++ b/vorlesungen/slides/8/chroma1.tex @@ -0,0 +1,56 @@ +% +% chroma1.tex -- slide template +% +% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule +% +\bgroup +\begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} +\frametitle{Schranke für $\operatorname{chr}(G)$} +\vspace{-20pt} +\begin{columns}[t,onlytextwidth] +\begin{column}{0.40\textwidth} +\begin{block}{Proposition} +Ist $G$ ein Graph mit maximalem Grad $d$, dann gilt +\[ +\operatorname{chr}(G) \le d + 1 +\] +\end{block} +\uncover<2->{% +\begin{block}{Beispiel} +\begin{itemize} +\item<3-> +Peterson-Graph $G$: maximaler Grad ist $d=3$, aber +\[ +\operatorname{chr}(G) += +3 +< d+1=4 +\] +\item<4-> +Voller Graph $V$: maximaler Grad ist $d=n-1$, +\[ +\operatorname{chr}(V) = n = d+1 +\] +\end{itemize} +\end{block}} +\end{column} +\begin{column}{0.58\textwidth} +\uncover<4->{% +\begin{proof}[Beweis] +Mit vollständiger Induktion, d.~h.~Annahme: Graphen mit $<n$ Knoten und +maximalem Grad $d$ lassen sich mit höchstens $d+1$ Farben färben. +\begin{itemize} +\item<5-> $X$ ein Graph mit $n$ Knoten +\item<6-> entferne den Knoten $v\in X$, $X'=X\setminus\{v\}$ +\item<7-> $X'$ lässt sich mit höchstens $d+1$ Farben einfärben +\item<8-> $v$ hat höchstens $d$ Nachbarn, die höchsten $d$ verschiedene +Farben haben +\item<9-> Es bleibt eine Farbe für $v$ +\end{itemize} +\end{proof}} +\end{column} +\end{columns} +\end{frame} +\egroup diff --git a/vorlesungen/slides/8/spanningtree.tex b/vorlesungen/slides/8/spanningtree.tex index 425fe1c..62180d9 100644 --- a/vorlesungen/slides/8/spanningtree.tex +++ b/vorlesungen/slides/8/spanningtree.tex @@ -3,6 +3,7 @@ % % (c) 2019 Prof Dr Andreas Müller, Hochschule Rapperswil % +\bgroup \begin{frame} \frametitle{Spannbäume} @@ -121,7 +122,7 @@ Wieviele Spannbäume gibt es? \begin{column}{0.56\hsize} \uncover<5->{% \begin{block}{Laplace-Matrix} -\vspace{-15pt} +%\vspace{-15pt} \[ L= \tiny @@ -162,3 +163,4 @@ L\text{ ohne }\left\{\begin{array}{c}\text{Zeile $i$}\\\text{Spalte $j$}\end{arr \end{columns} \end{frame} +\egroup |