The Number of Cycles of Bi-regular Tanner Graphs in Terms of the Eigenvalues of the Adjacency Matrix
Roxana Smarandache, David G. M. Mitchell
TL;DR
The paper addresses the problem of counting short cycles in Tanner graphs of biregular and QC-LDPC codes by linking cycle counts $N_{2k}$ to the eigenvalues of the adjacency matrix $A$. It develops a simple recursive formula for $N_{2k}$ in terms of the eigenvalues $\lambda$ of $HH^T$ (through $\alpha=\lambda^2-(q_1+q_2)$) and Newton's identities, enabling computation without explicit $A_e$ spectra. It further provides explicit expressions for $N_{2k}$ with $2\le k\le 7$ in terms of $\sum\lambda^k$ and relates these to a recursive, $E_k$-based formulation in terms of $\alpha$ as well as a variant in terms of $N_{2j}$ for $j<k$. The results offer efficient, exact cycle distribution calculations for QC-LDPC Tanner graphs, with practical impact on code design, performance analysis, and girth certification.
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.