Let G be a graph on n vertices, A its adjacency matrix (as described in the Graph Theory handout), and let d greaterthanorequalto 0 be an integer. Show that the ij^th entry in A^d is the number of walks in G of length d from vertex i to vertex j. Use the result of the preceding problem to prove the following. The distance from vertex i to vertex j is equal to min{ d | (A^d)_ij > 0}. Here (A^d)_ij denotes the ij^th entry in the matrix A^d