Detecting cycles in an adjacency matrix

Question

Let A be the adjacency matrix for the graph G = (V,E). A(i,j) = 1 if the nodes i and j are connected with an

Answer 1

Some more thoughts on the matrix approach... The example cited is the adjacency matrix for a disconnected graph (nodes 1&2 are connected, and nodes 3&4 are connected, but neither pair is connected to the other pair). When you calculate A^2, the answer (as stated) is the identity matrix. However, since Trace(A^2) = 4, this indicates that there are 2 loops each of length 2 (which is correct). Calculating A^3 is not permitted until these loops are properly identified and removed from the matrix. This is an involved procedure requiring several steps and is detailed nicely by R.L. Norman, "A Matrix Method for Location of Cycles of a Directed Graph," AIChE J, 11-3 (1965) pp. 450-452. Please note: it is unclear from the author whether this approach is guaranteed to find ALL cycles, UNIQUE cycles, and/or ELEMENTARY cycles. My experience suggests that it definitely does not identify ONLY unique cycles.