On Computing the Multiplicity of Cycles in Bipartite Graphs Using the Degree Distribution and the Spectrum of the Graph

TL;DR

This paper extends existing spectral methods to count various cycle lengths in bipartite graphs, including irregular and half-regular types, which is valuable for analyzing LDPC codes.

Contribution

It generalizes prior work by Blake and Lin to compute cycle counts of lengths beyond the girth in bipartite graphs, including irregular and half-regular cases.

Findings

Extended cycle counting to lengths g+2 to 2g-2 in bi-regular graphs

Counted 4-cycles and 6-cycles in irregular and half-regular graphs

Provided formulas based on degree distribution and spectrum

Abstract

Counting short cycles in bipartite graphs is a fundamental problem of interest in the analysis and design of low-density parity-check (LDPC) codes. The vast majority of research in this area is focused on algorithmic techniques. Most recently, Blake and Lin proposed a computational technique to count the number of cycles of length $g$ in a bi-regular bipartite graph, where $g$ is the girth of the graph. The information required for the computation is the node degree and the multiplicity of the nodes on both sides of the partition, as well as the eigenvalues of the adjacency matrix of the graph (graph spectrum). In this paper, the result of Blake and Lin is extended to compute the number of cycles of length $g+2, \ldots, 2g-2$, for bi-regular bipartite graphs, as well as the number of $4$-cycles and $6$-cycles in irregular and half-regular bipartite graphs, with $g \geq 4$ and $g \geq 6$, respectively.