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arXiv · 2601.05340

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

Roxana Smarandache·David G. M. Mitchell
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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.

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Roxana Smarandache, David G. M. Mitchell. 2026-01-08. The Number of Cycles of Bi-regular Tanner Graphs in Terms of the Eigenvalues of the Adjacency Matrix. https://arxiv.org/abs/2601.05340

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