dorsal/arxiv
View SchemaThe Number of Cycles of Bi-regular Tanner Graphs in Terms of the Eigenvalues of the Adjacency Matrix
| Authors | Roxana Smarandache, David G. M. Mitchell |
|---|---|
| Categories | |
| ArXiv ID | 2601.05340vv1 |
| URL | https://arxiv.org/abs/2601.05340 |
| License | http://creativecommons.org/licenses/by-nc-sa/4.0/ |
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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"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 $2k$, $k\u003cg$, 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.",
"arxiv_id": "2601.05340",
"authors": [
"Roxana Smarandache",
"David G. M. Mitchell"
],
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"cs.IT",
"math.IT"
],
"license": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
"title": "The Number of Cycles of Bi-regular Tanner Graphs in Terms of the Eigenvalues of the Adjacency Matrix",
"url": "https://arxiv.org/abs/2601.05340",
"version": "v1"
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