The Number of Cycles of Bi-regular Tanner Graphs in Terms of the Eigenvalues of the Adjacency Matrix

TL;DR

This paper establishes new relationships between cycles in LDPC code graphs and adjacency matrix eigenvalues, providing formulas to count cycles efficiently, especially for bi-regular QC-LDPC codes.

Contribution

It introduces recursive and explicit formulas linking cycle counts to eigenvalues, enhancing analysis of LDPC code graphs with practical computational methods.

Findings

Derived recursive formulas for cycle counts in bi-regular graphs.

Explicit formulas for cycle counts up to length 14 in terms of eigenvalues.

Applicable to efficiently analyze QC-LDPC codes.

Abstract

In this paper, we explore new connections between the cycles in the graph of low-density parity-check (LDPC) codes and the eigenvalues of the corresponding adjacency matrix. The resulting observations are used to derive fast, simple, recursive formulas for the number of cycles $N_{2k}$ of length $2k$, $k<g$, in a bi-regular graph of girth $g$. Moreover, we derive explicit formulas for $N_{2k}$, $k\leq 7$, in terms of the nonzero eigenvalues of the adjacency matrix. Throughout, we focus on the practically interesting class of bi-regular quasi-cyclic LDPC (QC-LDPC) codes, for which the eigenvalues can be obtained efficiently by applying techniques used for block-circulant matrices.