Exam 10: Graphs

In questions fill in the blanks.1113:1156 -If $G$ is a connected graph with 12 regions and 20 edges, then $G$ has $\underline{\quad\quad}$ vertices.

for each graph give an ordered pair description (vertex set and edge set) and an adjacency matrix, and draw a picture of the graph. -K6.

Solve the traveling salesman problem for the graph at the right by finding the total weight of all Hamilton circuits and determining a circuit with minimum total weight. Inof questionsstreets $( m \leq n )$

In $K _ { 5 }$ find the number of paths of length 3 between every pair of vertices.

Are these two graphs isomorphic?

Suppose G is a graph with vertices a, b, c, d, e, f with adjacency matrix $\left( \begin{array} { l l l l l l } 0 & 1 & 0 & 1 & 0 & 0 \\1 & 0 & 0 & 1 & 1 & 1 \\0 & 0 & 0 & 0 & 1 & 1 \\1 & 1 & 0 & 0 & 1 & 0 \\0 & 1 & 1 & 1 & 0 & 1 \\0 & 1 & 1 & 0 & 1 & 0\end{array} \right)$ (where alpha- betical order is used to determine the rows and columns of the adjacency matrix). Find (a) the number of vertices in G. (b) the number of edges in G. (c) the degree of each vertex. (d) the number of loops. (e) the length of the longest simple path in G. (f) the number of components in G. (g) the distance between vertex a and vertex c. Questions 94-96 refer to a cubic graph, i.e., a graph that is simple and has every vertex of degree 3.

Consider the graph at the right. (a) Does it have an Euler circuit? (b) Does it have an Euler path? (c) Does it have a Hamilton circuit? (d) Does it have a Hamilton path?