2018-11-27 15:14:50 +01:00
|
|
|
\documentclass[12pt,a4paper,german]{article}
|
|
|
|
|
\usepackage{url}
|
|
|
|
|
%\usepackage{graphics}
|
|
|
|
|
\usepackage{times}
|
|
|
|
|
\usepackage[T1]{fontenc}
|
|
|
|
|
\usepackage{ngerman}
|
|
|
|
|
\usepackage{float}
|
|
|
|
|
\usepackage{diagbox}
|
|
|
|
|
\usepackage[utf8]{inputenc}
|
|
|
|
|
\usepackage{geometry}
|
|
|
|
|
\usepackage{amsfonts}
|
|
|
|
|
\usepackage{amsmath}
|
|
|
|
|
\usepackage{cancel}
|
|
|
|
|
\usepackage{csquotes}
|
|
|
|
|
\usepackage{graphicx}
|
|
|
|
|
\usepackage{epsfig}
|
|
|
|
|
\usepackage{paralist}
|
|
|
|
|
\usepackage{tikz}
|
|
|
|
|
\geometry{left=2.0cm,textwidth=17cm,top=3.5cm,textheight=23cm}
|
|
|
|
|
|
|
|
|
|
%%%%%%%%%% Fill out the the definitions %%%%%%%%%
|
|
|
|
|
\def \name {Valentin Brandl} %
|
|
|
|
|
\def \matrikel {108018274494} %
|
|
|
|
|
\def \pname {Marvin Herrmann} %
|
|
|
|
|
\def \pmatrikel {108018265436} %
|
|
|
|
|
\def \gruppe {2 (Mi 10-12 Andre)}
|
|
|
|
|
\def \qname {Pascal Brackmann}
|
|
|
|
|
\def \qmatrikel {108017113834} %
|
|
|
|
|
\def \uebung {6} %
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
|
|
|
|
|
|
% DO NOT MODIFY THIS HEADER
|
|
|
|
|
\newcommand{\hwsol}{
|
|
|
|
|
\vspace*{-2cm}
|
|
|
|
|
\noindent \matrikel \quad \name \hfill \"Ubungsgruppe: \gruppe \\
|
|
|
|
|
\noindent \pmatrikel \quad \pname \\
|
|
|
|
|
\noindent \qmatrikel \quad \qname \\
|
|
|
|
|
\begin{center}{\Large \bf L\"osung f\"ur \"Ubung \# \uebung}\end{center}
|
|
|
|
|
}
|
|
|
|
|
|
|
|
|
|
\begin{document}
|
|
|
|
|
%Import header
|
|
|
|
|
\hwsol
|
|
|
|
|
|
|
|
|
|
\section*{Aufgabe 6.3}
|
|
|
|
|
|
|
|
|
|
\begin{enumerate}[1.]
|
|
|
|
|
|
|
|
|
|
\item Anzahl der Automorphismen des vollständigen Graphen $K_n = (V,E)$ mit $V = \{1,...,n\}$ und
|
|
|
|
|
$E = \{\{i,j\} | 1 \leq i,j \leq n \land i \neq j \}$
|
|
|
|
|
|
|
|
|
|
\item Anzahl der Automorphismen des Sterngraphen $S_n = (V,E)$ mit $V = \{1,...,n\}$ und
|
|
|
|
|
$E = \{\{1,i\} | i = 2,...n\}$
|
|
|
|
|
|
|
|
|
|
Der innere Knoten $i = 1$ kann sich nicht verändern, da er der einziger Knoten mit $deg(i) = n-1 \neq 1$ ist.
|
|
|
|
|
|
|
|
|
|
Die äußeren Knoten $O = V\textbackslash{}\{i\}$ können beliebig getauscht werden, da sie nur je einen
|
|
|
|
|
gemeinsamen Nachbarn haben: $\forall v \in O, \Gamma(v) = \{i\} \land deg(v) = 1$
|
|
|
|
|
|
|
|
|
|
$\Rightarrow (n-1)!$ Automorphismen
|
|
|
|
|
|
|
|
|
|
\item Assymetrischer Graph mit Knotenanzahl $n = 6$
|
|
|
|
|
|
2018-11-27 15:24:58 +01:00
|
|
|
\includegraphics[scale=0.5]{./build/school/di-ma/uebung/06/06_3.png}
|
|
|
|
|
|
2018-11-27 15:14:50 +01:00
|
|
|
\end{enumerate}
|
|
|
|
|
|
|
|
|
|
\end{document}
|
