Counting short cycles in bipartite graphs is a fundamental problem of interest in many fields including the analysis and design of low-density parity-check (LDPC) codes. There are two computational approaches to count short cycles (with length smaller than $2g$ , where $g$ is the girth of the graph) in bipartite graphs. The first approach is applicable to a general (irregular) bipartite graph, and uses the spectrum $\{\eta _{i}\}$ of the directed edge matrix of the graph to compute the multiplicity $N_{k}$ of $k$ -cycles with $k < 2g$ through the simple equation $N_{k} = \sum _{i} \eta _{i}^{k}/(2k)$ . This approach has a computational complexity $\mathcal {O}(|E|^{3})$ , where $|E|$ is number of edges in the graph. The second approach is only applicable to bi-regular bipartite graphs, and uses the spectrum $\{\lambda _{i}\}$ of the adjacency matrix (graph spectrum) and the degree sequences of the graph to compute $N_{k}$ . The complexity of this approach is $\mathcal {O}(|V|^{3})$ , where $|V|$ is number of nodes in the graph. This complexity is less than that of the first approach, but the equations involved in the computations of the second approach are complex and tedious, particularly for $k \geq g+6$ . In fact, the computational complexity of the equations increases exponentially with $k$ . In this paper, we establish an analytical relationship between the two spectra $\{\eta _{i}\}$ and $\{\lambda _{i}\}$ for bi-regular bipartite graphs. Through this relationship, the former spectrum can be derived from the latter through simple equations with computational complexity constant in $k$ . This allows the computation of $N_{k}$ using $N_{k} = \sum _{i} \eta _{i}^{k}/(2k)$ but with a complexity of $\mathcal {O}(|V|^{3})$ rather than $\mathcal {O}(|E|^{3})$ .

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关于使用有向边矩阵的谱计算二部图中的短环数

计算二部图中的短周期是许多领域中的一个基本问题，包括低密度奇偶校验 (LDPC) 码的分析和设计。有两种计算短周期（长度小于 $2g$ ， 在哪里 $g$ 是图的周长）在二部图中。第一种方法适用于一般（不规则）二部图，并使用频谱 $\{\eta _{i}\}$ 图的有向边矩阵的计算多重性 $N_{k}$ 的 $千$ - 循环 $k < 2g$ 通过简单的方程 $N_{k} = \sum _{i} \eta _{i}^{k}/(2k)$ . 这种方法具有计算复杂度 $\mathcal {O}(|E|^{3})$ ， 在哪里 $|E|$ 是图中的边数。第二种方法只适用于双正则二部图，并且使用谱 $\{\lambda _{i}\}$ 邻接矩阵（图谱）和要计算的图的度数序列 $N_{k}$ . 这种方法的复杂性是 $\mathcal {O}(|V|^{3})$ ， 在哪里 $|V|$ 是图中的节点数。这种复杂度低于第一种方法，但第二种方法计算中涉及的方程复杂而乏味，特别是对于 $k \geq g+6$ . 事实上，方程的计算复杂度随着 $千$ . 在本文中，我们建立了两个光谱之间的解析关系 $\{\eta _{i}\}$ 和 $\{\lambda _{i}\}$ 对于双正则二部图。通过这种关系，可以通过计算复杂度为常数的简单方程从后者导出前者的频谱 $千$ . 这允许计算 $N_{k}$ 使用 $N_{k} = \sum _{i} \eta _{i}^{k}/(2k)$ 但具有复杂性 $\mathcal {O}(|V|^{3})$ 而不是 $\mathcal {O}(|E|^{3})$ .