If A is the adjacenty matrix associated with a simple graph, then the (i,j) entry in A*A*A=A^3 is the number of routes of lengths 1 or 2 from node i to node j the number of paths of length exactly 3 from node i to node j the number of paths of lengths 1 or 2 from node i to node j the number of paths of not more than 3 edges from node i to node j the number of routes of not more than 3 edges from node i to node j (the number of routes of length exactly 3 from node i to node i "the number of edges joining nodes i and j