On short cycle enumeration in biregular bipartite graphs

TL;DR

This paper presents a method to compute the number of short cycles, including the girth, in biregular bipartite graphs used for LDPC codes, based solely on their parameters, simplifying analysis of their error floors.

Contribution

It provides a graph-theoretic approach to directly compute girth and short cycle counts in biregular bipartite graphs from basic parameters, reducing computational complexity.

Findings

Girth and short cycle counts can be computed from graph parameters

The method simplifies analysis of LDPC Tanner graphs

Reduces reliance on complex algorithms for cycle counting

Abstract

A number of recent works have used a variety of combinatorial constructions to derive Tanner graphs for LDPC codes and some of these have been shown to perform well in terms of their probability of error curves 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 here. One property 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.