280 Solutions to Exercises (b) A 3 is the adjacency matrix of a graph 9 ~ the connected components of 9 are of the form 0 or 0-0. Proof. Suppose that A3 is the adjacency matrix of a graph 9. Then all entries of A3 must be 0 or 1. If 9 contains any vertex of degree d > 1, say v, then we have w-v-x. But now, there are two walks oflength 3 from v to w; namely, vxvw and vwvw, so the (v, w) entry of A3 would be bigger than 1. We conclude that every vertex of 9 is of degree 0 or 1; that is, the connected components of 9 are of the form 0 or 0-0. Conversely, if 9 is a graph whose connected components are all of this form, then all entries of A3 are 0 or 1 (aij = 1 ~ ViO-OVj). Note that the diagonal entries are all 0 since there are no triangles. Hence, A 3 is the adjacency matrix of some graph. • 16. (----+) Suppose the graph is connected. Then there is a walk between every pair of vertices. Let n be the length of the longest such walk. Then for any i and j, there is a walk of length k ::; n from i to j, and so the (i,j) entry of Ak is 1.
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