from the above exampleswe can draw the following conclusionsfor simple graph (that has no loops)the adjacency matrix has on the diagonal the adjacency matrix of an undirected graph is symmetric the memory use of an adjacency matrix is ( )where is the number of nodes in the graph number of (or non-zero entriesin an adjacency matrix is equal to the number of edges in the graph the adjacency matrix for weighted graph contains the weights of the edges connecting the nodes now let us discuss the powers of an adjacency matrix from adjacency matrix we can conclude that an entry in the ith row and jth column means that there exists path of length from vi to vj now considera and (aij) aaik akj any entry aij if aik akj that isif there is an edge (vivkand (vkvj)then there is path from vi to vj of length similarlyevery entry in the ith row and jth column of gives the bcd number of paths of length from node vi to vj in general termswe can conclude that every entry in the ith row and jth column of an (where is the number of nodes in the graphgives the number of paths of length from node vi to vj consider directed figure directed graph graph given in fig given its adjacency matrix alet us calculate with its adjacency and matrix ya ii ii ii ii ii ii nowbased on the above calculationswe define matrix asbr ar an entry in the ith row and jth column of matrix br gives the number of paths of length or less than from vertex vi to vj the main goal to define matrix is to obtain the path matrix the path matrix can be calculated from by setting an entry pij if bij is non-zero and pij