Chapter 1, Problem 6E

Consider the graph

(a) Determine the adjacency matrix A for the graph and enter it in MATLAB.
(b) Compute $A^2$ and determine the number of walks of length 2 from (i) $V_1$ to $V_7$ (ii) $V_4$ to $V_8$ (iii) $V_5$ to $V_6$ , and (iv) $V_8$ to $V_3$ .
(c) Compute $A^4,A^6$ , and $A^8$ and answer the questions in part (b) for walks of lengths 4, 6, and 8.
Make a conjecture as to when there will be no walks of even length from vertex $V_i$ to vertex $V_j$ .
(d) Compute $A^3,A^5$ , and $A^7$ and answer the questions from part (b) for walks of lengths 3, 5, and 7. Does your conjecture from part (c) hold for walks of odd length? Explain. Make a conjecture as to whether there are any walks of length k from $V_i$ to $V_j$ , based on whether $i+j+k$ is odd or even.
(e) If we add the edges $\left\{V_3,V_6\right\},\left\{V_5,V_8\right\}$ to the graph, the adjacency matrix B for the new graph can be generated by setting $B=A$ and then setting
$\begin{array}{cc}B\left(3,6\right)=1,& B\left(6,3\right)=1,\\ B\left(5,8\right)=1,& B\left(8,5\right)=1\end{array}$
Compute $B^k$ for $k=2,3,4,5$ . Is your conjecture from part (d) still valid for the new graph?
(f) Add the edge $\left\{V_6,V_8\right\}$ to the figure and construct the adjacency matrix C for the resulting graph. Compute powers of C to determine whether your conjecture from part (d) will shit hold for this new graph.