91Ó°ÊÓ

Consider the graph with vertex set \(\\{A, B, C, D, E\\}\) and edge list \(A C, A E, B D, B E, C A, C D, C E, D E .\) Draw two different pictures of the graph.

Consider the graph with vertex set \(\\{A, B, C, D, E\\}\) and edge list \(A D, A E, B C, B D, D D, D E .\) Without drawing a picture of the graph (a) list all the vertices adjacent to \(D\). (b) list all the edges adjacent to \(B D\). (c) find the degree of \(D\). (d) find the sum of the degrees of the vertices.

Table 3 summarizes the Facebook friendships between a group of eight individuals [an \(F\) indicates that the individuals (row and column) are Facebook friends]. Draw a graph that models the set of friendships in the group. (Use the first letter of the name to label the vertices.) $$\begin{array}{|l|c|c|c|c|c|c|c|c|}\hline & \text { Fred } & \text { Pat } & \text { Mac } & \text { Ben } & \text { Tom } & \text { Hale } & \text { Zac } & \text { Cher } \\\\\hline \text { Fred } & & \mathrm{F} & & & \mathrm{F} & \mathrm{F} & & \\\\\hline \text { Pat } & \mathrm{F} & & & & \mathrm{F} & \mathrm{F} & & \mathrm{F} \\\\\hline \text { Mac } & & & & \mathrm{F} & & & \mathrm{F} & \\\\\hline \text { Ben } & & & \mathrm{F} & & & & \mathrm{F} & \\\\\hline \text { Tom } & \mathrm{F} & \mathrm{F} & & & & \mathrm{F} & & \\ \hline \text { Hale } & \mathrm{F} & \mathrm{F} & & & \mathrm{F} & & & \mathrm{F} \\\\\hline \text { Zac } & & & \mathrm{F} & \mathrm{F} & & & & \\\\\hline \text { Cher } & & \mathrm{F} & & & & \mathrm{F} & & \\\\\hline\end{array}$$

(a) Explain why in every graph the sum of the degrees of all the vertices equals twice the number of edges. (b) Explain why every graph must have an even number of odd vertices.