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A message-passing algorithm for counting short cycles in a graph ispresented. For bipartite graphs, which are of particular interest in coding,the algorithm is capable of counting cycles of length g, g +2,..., 2g - 2,where g is the girth of the graph. For a general (non-bipartite) graph, cyclesof length g; g + 1, ..., 2g - 1 can be counted. The algorithm is based onperforming integer additions and subtractions in the nodes of the graph andpassing extrinsic messages to adjacent nodes. The complexity of the proposedalgorithm grows as $O(g|E|^2)$, where $|E|$ is the number of edges in thegraph. For sparse graphs, the proposed algorithm significantly outperforms theexisting algorithms in terms of computational complexity and memoryrequirements.

A Message-Passing Algorithm for Counting Short Cycles in a Graph