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

2026-01-08

arXiv.org

AI Summary

Efficiently counting short cycles in biregular Tanner graphs is crucial for analyzing the performance of LDPC codes. This work presents the first formulation of this problem as a function of the non-zero eigenvalues of the adjacency matrix, leveraging tools from graph theory and spectral analysis of block-circulant matrices. The authors derive a compact recursive formula applicable to all even-length cycles up to girth \( g \), and provide explicit closed-form expressions specifically for biregular quasi-cyclic LDPC codes. The proposed method significantly improves computational efficiency, enabling the derivation of closed-form solutions for cycle counts of length up to 14. These results offer a practical tool for efficient cycle detection and code design in LDPC systems.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• cycles
• bi-regular Tanner graphs
• eigenvalues
• LDPC codes
• adjacency matrix

Innovation

• bi-regular Tanner graphs
• cycle enumeration
• eigenvalues of adjacency matrix
• QC-LDPC codes
• recursive formulas