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.

翻译：本文探讨了低密度奇偶校验(LDPC)码的图结构中环与对应邻接矩阵特征值之间的新联系。基于所得观察结果，我们推导出围长为g的双正则图中长度为2k（k<g）的环数量N_{2k}的快速、简洁的递归计算公式。进一步地，我们根据邻接矩阵的非零特征值，给出了k≤7时N_{2k}的显式表达式。全文聚焦于具有实际应用价值的双正则准循环LDPC(QC-LDPC)码类，其邻接矩阵特征值可通过分块循环矩阵的相关技术高效计算。