On Short Cycle Enumeration in Biregular Bipartite Graphs

Abstract

A number of recent works have used a variety of combinatorial constructions to derive Tanner graphs for LDPC codes and some of these LDPC codes have been shown to perform well in terms of their probability of errors and error floors. Such graphs are bipartite and many of these constructions yield biregular graphs, where the degree of left vertices is a constant $c+1$ and that of the right vertices is a constant $d+1$ . Such graphs are termed $(c+1,d+1)$ biregular bipartite graphs. Two properties of interest in such work is the girth of the graph and the number of short cycles in the graph, cycles of length either the girth or slightly larger. Such numbers have been shown to be related to the error floor of the probability of error curve of the related LDPC code. Using known results of graph theory, it is shown how the girth and the number of cycles of length equal to the girth may be computed for these $(c+1, d+1)$ biregular bipartite graphs knowing only the parameters $c$ and $d$ and the numbers of left and right vertices. While numerous algorithms to determine the number of short cycles in arbitrary graphs exist, the reduction of the problem from an algorithm to a computation for these biregular bipartite graphs is of interest.