The Number of Fixed Length Cycles in Undirected Graph Explicit Formula in Case of Small Lengths

Modifications of Ross and Harary algorithm to express the number ck of cycles of length k in an undirected graph in terms of its adjacency matrix are developed. The general undirected graphs as well as bipartite graphs were considered. Computer algebra implementations of the algorithms enable us to construct the formulae at least for k ≤ 12 in general case and for k ≤ 14 in case of bipartite graph. It was shown that, for any fixed value of k ≥ 8 and space complexity quadratic in order n of a graph, the time complexity of computing ck is O(n[k/2] logn). In case of bipartite graph, for k = 8,10,14 better estimations are obtained: O(n3 log2n), O(n4 log2n), O(n6 log2n).

graph algorithms, cycles in graph, adjacency matrix