From the Spectrum of the Adjacency Matrix to the Spectrum of Directed Edge Matrix: Counting Cycles of a Bipartite Graph Through a Simple Equation

Abstract

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 Nk of k-cycles with $k \lt 2g$ through the simple equation $N_{k} =\sum_{i}\eta_{i}^{k}/(2k)$. This approach has a computational complexity $O(|E|^{3})$, 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 Nk. The complexity of this approach is $O(|V|^{3})$, 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 very tedious, particularly for $k \geq g+6$. 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. This allows the computation of Nk using $N_{k} =\sum_{i}\eta_{i}^{k}/(2k)$ but with a complexity of $O(|V|^{3})$ rather than $O(|E|^{3})$.