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

Abstract

Recently, Tanner graphs which represented low density parity check (LDPC) codes have become an interesting research topic. Finding the number of short cycles of Tanner graphs motivated Blake and Lin to investigate the multiplicity of cycles of length equal to the girth of bi-regular bipartite graphs by using the spectrum and degree distribution of the graph. While there were many algorithms to find the number of cycles, they chose to take a computational approach. Dehghan and Banihashemi counted the number of cycles of length [Formula: see text] and [Formula: see text] where [Formula: see text] is a bi-regular bipartite graph and [Formula: see text] is the girth of [Formula: see text] But for the cycles of length smaller than [Formula: see text] in bi-regular bipartite graphs, they only proposed a descriptive technique. In this paper, we find the number of cycles of length less than [Formula: see text] by using the spectrum and the degree distribution of bi-regular bipartite graphs such that the formula depends only on the partitions of positive integers and the number of closed cycle-free walks from any vertex of [Formula: see text] and [Formula: see text] which are known.