On Computing the Number of Short Cycles in Bipartite Graphs Using the Spectrum of the Directed Edge Matrix

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 <inline-formula> <tex-math notation="LaTeX">$2g$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> is the girth of the graph) in bipartite graphs. The first approach is applicable to a general (irregular) bipartite graph, and uses the spectrum <inline-formula> <tex-math notation="LaTeX">$\{\eta _{i}\}$ </tex-math></inline-formula> of the directed edge matrix of the graph to compute the multiplicity <inline-formula> <tex-math notation="LaTeX">$N_{k}$ </tex-math></inline-formula> of <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>-cycles with <inline-formula> <tex-math notation="LaTeX">$k < 2g$ </tex-math></inline-formula> through the simple equation <inline-formula> <tex-math notation="LaTeX">$N_{k} = \sum _{i} \eta _{i}^{k}/(2k)$ </tex-math></inline-formula>. This approach has a computational complexity <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(|E|^{3})$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$|E|$ </tex-math></inline-formula> is number of edges in the graph. The second approach is only applicable to bi-regular bipartite graphs, and uses the spectrum <inline-formula> <tex-math notation="LaTeX">$\{\lambda _{i}\}$ </tex-math></inline-formula> of the adjacency matrix (graph spectrum) and the degree sequences of the graph to compute <inline-formula> <tex-math notation="LaTeX">$N_{k}$ </tex-math></inline-formula>. The complexity of this approach is <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(|V|^{3})$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">$|V|$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$k \geq g+6$ </tex-math></inline-formula>. In fact, the computational complexity of the equations increases exponentially with <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>. In this paper, we establish an analytical relationship between the two spectra <inline-formula> <tex-math notation="LaTeX">$\{\eta _{i}\}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\{\lambda _{i}\}$ </tex-math></inline-formula> 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 <inline-formula> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula>. This allows the computation of <inline-formula> <tex-math notation="LaTeX">$N_{k}$ </tex-math></inline-formula> using <inline-formula> <tex-math notation="LaTeX">$N_{k} = \sum _{i} \eta _{i}^{k}/(2k)$ </tex-math></inline-formula> but with a complexity of <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(|V|^{3})$ </tex-math></inline-formula> rather than <inline-formula> <tex-math notation="LaTeX">$\mathcal {O}(|E|^{3})$ </tex-math></inline-formula>.