Introduction to Discrete Math

are non-zero. Solution: First suppose the above some has no non-zero entries on its diagonal. Since A(G)i,j is the number of walks of length k from i to j, if the sum above has non-zero diagonal entries, then for any pair of vertices i, j there is a walk of some length from i to j. This implies there is a path from i to j. Since i, j are arbitrary, we conclude that G is connected. Now assume G is connected. Then there is a path from any vertex i to any vertex j, of some length. Since the path can not repeat vertices, it has length at most n − 1, so there is some positive integer k less than n such that A(G)i,j = 1. Doing this for every pair i 6= j we get that the sum A(G) +A2(G) + · · ·+An(G) has non-zero entries off its diagonal. (b) Let Jn and In be the n×n all 1s matrix and n×n identity matrix, respectively. If A(G) satisfies