\section{Graphs}

\definition{}
A \textit{set} is an unordered collection of objects. \par
This means that the sets $\{1, 2, 3\}$ and $\{3, 2, 1\}$ are identical.


\definition{}
A \textit{graph} $G = (N, E)$ consists of two sets: a set of \textit{vertices} $V$, and a set of \textit{edges} $E$. \par
Vertices are simply named \say{points,} and edges are connections between pairs of vertices. \par
In the graph below, $V = \{a, b, c, d\}$ and $E = \{~ (a,b),~ (a,c),~ (a,d),~ (c,d) ~\}$.

\begin{center}
\begin{tikzpicture}
	\begin{scope}[layer = nodes]
		\node[main] (a) at (0, 0) {$a$};
		\node[main] (b) at (0, -1) {$b$};
		\node[main] (c) at (2, -1) {$c$};
		\node[main] (d) at (4, 0) {$d$};
	\end{scope}

	\draw[-]
		(a) edge (b)
		(a) edge (c)
		(a) edge (d)
		(c) edge (d)
	;
\end{tikzpicture}
\end{center}

Vertices are also sometimes called \textit{nodes}. You'll see both terms in this handout. \par



\problem{}
Draw the graph defined by the following vertex and edge sets: \par
$V = \{A,B,C,D,E\}$ \par
$E = \{~ (A,B),~ (A,C),~ (A,D),~ (A,E),~ (B,C),~ (C,D),~ (D,E) ~\}$\par

\vfill


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.
How did she know?


\vfill

\problem{}
Say $G$ is a graph with nine vertices. Show that $G$ has at least five vertices of degree six or at least six vertices of degree 5.

\vfill
\pagebreak