The Number of Cycles of Bi-regular Tanner Graphs in Terms of the Eigenvalues of the Adjacency Matrix
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914242485026816 |
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| author | Smarandache, Roxana Mitchell, David G. M. |
| author_facet | Smarandache, Roxana Mitchell, David G. M. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_05340 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Number of Cycles of Bi-regular Tanner Graphs in Terms of the Eigenvalues of the Adjacency Matrix Smarandache, Roxana Mitchell, David G. M. Information Theory 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. |
| title | The Number of Cycles of Bi-regular Tanner Graphs in Terms of the Eigenvalues of the Adjacency Matrix |
| topic | Information Theory |
| url | https://arxiv.org/abs/2601.05340 |