We can use graphs to solve many different kinds of problems. \par
Most situations that involve some kind of \say{relation} between elements can be represented by a graph.
\pagebreak
Graphs are fully defined by their vertices and edges. The exact position of each vertex and edge doesn't matter---only which nodes are connected to each other. The same graph can be drawn in many different ways.
\problem{}
Show that the graphs below are equivalent by comparing the sets of their vertices and edges.
\begin{center}
\adjustbox{valign=c}{
\begin{tikzpicture}
\begin{scope}[layer = nodes]
\node[main] (a) at (0, 0) {$a$};
\node[main] (b) at (2, 0) {$b$};
\node[main] (c) at (2, -2) {$c$};
\node[main] (d) at (0, -2) {$d$};
\end{scope}
\draw[-]
(a) edge (b)
(b) edge (c)
(c) edge (d)
(d) edge (a)
(a) edge (c)
(b) edge (d)
;
\end{tikzpicture}
}
\hspace{20mm}
\adjustbox{valign=c}{
\begin{tikzpicture}
\begin{scope}[layer = nodes]
\node[main] (a) at (0, 0) {$a$};
\node[main] (b) at (-2, -2) {$b$};
\node[main] (c) at (0, -2) {$c$};
\node[main] (d) at (2, -2) {$d$};
\end{scope}
\draw[-]
(a) edge (b)
(b) edge (c)
(c) edge (d)
(d) edge (a)
(a) edge (c)
(b) edge[out=270, in=270, looseness=1] (d)
;
\end{tikzpicture}
}
\end{center}
\vfill
\pagebreak
\definition{}
The degree $D(v)$ of a vertex $v$ of a graph
is the number of the edges of the graph
connected to that vertex.
\theorem{Handshake Lemma}<handshake>
In any graph, the sum of the degrees of its vertices equals twice the number of the edges.
\problem{}
Prove \ref{handshake}.
\vfill
\problem{}
Show that all graphs have an even number number of vertices with odd degree.
\vfill
\problem{}
One girl tells another, \say{There are 25 kids
in my class. Isn't it funny that each of them
has 5 friends in the class?}\say{This cannot be true,} immediately replies the other girl.
