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AND \section{GRAPAHS SUBGRAPHS}
\subsection{Graphs and Simple Graphs}
\begin{itemize}
\item
List five situations from everyday life in which graphs arise naturally.
\item
Draw a different diagram of the graph of figure 1.3a to show that it is indeed planar.
\item
Show that if G is simple, then $\varepsilon \leqslant \binom{\nu}{2}$.
\end{itemize}
\subsection{Graph Isomorphism}
\begin{itemize}
\item
Find an isomorphism between the graphs $G$ and $H$ of examples 1 and 2 different from the one given.
\item
(a) Show that if $G \cong H$, then $\nu(G) = \nu(H)$ and $\varepsilon(G) = \varepsilon(H)$.
(b) Give an example to show that the converse is false.\label{ex1_2_5}
\item
Show that the following graphs are isomorphic:
\item
Show that there are eleven nonisomorphic simple graphs on four vertices.
\item
Show that two simple graphs $G$ and $H$ are isomorphic if and only if there is a bijection $\theta: V(G) \to V(H)$ such that $uv \in E(G)$ if and only if $\theta(u)\theta(v)\in E(H)$.
\item Show that the following graphs are isomorphic:
\item
Let $G$ be simple. Show that $\varepsilon=\binom{\nu}{2}$ if and only if $G$ is complete.
\item
Show that
(a) $\varepsilon(K_{m,n})=mn$;
(b) if $G$ is simple and bipartite, then $\varepsilon \leqslant \nu^2/4$.
\item
A \emph{$k$-partite graph} is one whose vertex set can be partitioned into
$k$ subsets so that no edge has both ends in any one subset;
a \emph{complete $k$-partite graph} is one that is simple and in which each vertex is joined to every vertex that is not in the same subset.
The complete $m$-partite graph on $n$ vertices in which each part has either $[n/m]$ or $\{n/m\}$ vertices is denoted by $T_{m,n}$. Show that
(a) $\varepsilon(T_{m,n})=\binom{n-k}{2}+(m-1)\binom{k+1}{2}$, where $k=[n/m]$;
(b) if $G$ is a complete $m$-partite graph on $n$ vertices, then $\varepsilon(G) \leqslant \varepsilon(T_{m,n})$, with equality only if $G \cong T_{m,n}$.
\item
The \emph{$k$-cube} is the graph whose vertices are the ordered of $k$-tuples 0's and 1's, two vertices being joined if and only if they differ in exactly one coordinate. (The graph shown in figure 1.4b is just the 3-cube.) Show that the $k$-cube has $2^k$ vertices, $k2^{k-1}$ edges and is bipartite.
\item
(a) The \emph{complement} $G^c$ of a simple graph $G$ is the simple graph with vertex set $V$, two vertices being adjacent in $G^c$ if and only if they are not adjacent in $G$. Describe the graphs $K_n^c$ and $K_{m,n}^c$.
(b) A simple graph $G$ is \emph{self-complementary} if $G\cong G^c$. Show that if $G$ is self-complementary, then $\nu \equiv 0,\;1\pmod 4 $.
\item
An \emph{automorphism} of a graph is an isomorphism of the graph onto itself.
(a) Show, using exercise \ref{ex1_2_5}, that an automorphism of a simple graph $G$ can be regarded as a permutation on $V$ which preserves adjacency, and that the set of such permutations form a group $\Gamma(G)$ (the \emph{automorphism group} of $G$) under the usual operation of composition.
(b) Find $\Gamma(K_n)$ and $\Gamma(K_{m,n})$.
(c) Find a nontrivial simple graph whose automorphism group is the identity.
(d) Show that for any simple graph $G$, $\Gamma(G)=\Gamma(G^c)$.
(e) Consider the permutation group $\Lambda$ with elements $(1)(2)(3)$, $(1,2,3)$ and $(1,3,2)$. Show that there is no simple graph $G$ with vertex set $\{1,2,3\}$ such that $\Gamma(G)=\Lambda$.
(f) Find a simple graph $G$ such that $\Gamma(G) = \Lambda$. (Frucht, 1939 has shown that every abstract group is isomorphic to the automorphism group of some graph.)
\item
A simple graph $G$ is \emph{vertex-transitive} if, for any two vertices $u$ and $v$,
there is an element $g$ in $\Gamma(G)$ such that $g(u)=g(v)$;
$G$ is \emph{edge-transitive} if, for any two edges $u_1 v_1$ and $u_2 v_2$, there is an element $h$ in $\Gamma(G)$ such that $h(\{u_1, v_1\})=\{u_2, v_2\}$. Find
(a) a graph which is vertex-transitive but not edge-transitive;
(b) a graph which is edge-transitive but not vertex-transitive.
\end{itemize}

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