Counting short cycles of (c,d)-regular bipartite graphs

TL;DR

This paper presents a new method to count short cycles in bi-regular bipartite graphs, which are crucial for analyzing Tanner graphs in LDPC codes, using spectral properties and degree distributions.

Contribution

It introduces a formula based on spectrum and degree distribution to count cycles shorter than twice the girth, extending previous descriptive techniques.

Findings

Provides a formula for counting cycles less than 2g in bi-regular bipartite graphs.

Uses spectrum and degree distribution to derive cycle counts.

Enhances understanding of cycle structures in Tanner graphs for LDPC codes.

Abstract

Recently, working on the Tanner graph which represents a low density parity check (LDPC) code becomes an interesting research subject. Finding the number of short cycles of Tanner graphs motivated Blake and Lin to investigate the multiplicity of cycles of length girth in bi-regular bipartite graphs, by using the spectrum and degree distribution of the graph. Although there were many algorithms to find the number of cycles, they preferred to investigate in a computational way. Dehghan and Banihashemi counted the number of cycles of length $g+2$ and $g+4,$ where $G$ is a bi-regular bipartite graph and $g$ is the length of the girth $G.$ But they just proposed a descriptive technique to compute the multiplicity of cycles of length less than $2g$ for bi-regular bipartite graphs. In this paper, we find the number of cycles of length less than $2g$ by using spectrum and degree distribution of a bi-regular bipartite graph such that the formula depends only on the partitions of positive integers and the number of closed cycle-free walks from a vertex of the bi-regular bipartite graph.