On Computing the Multiplicity of Cycles in Bipartite Graphs Using the Degree Distribution and the Spectrum of the Graph

Counting short cycles in bipartite graphs is a fundamental problem of interest in the analysis and design of low-density parity-check codes. The vast majority of research in this area is focused on algorithmic techniques. Most recently, Blake and Lin proposed a computational technique to count the number of cycles of length <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {g}$ </tex-math></inline-formula> in a bi-regular bipartite graph, where <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {g}$ </tex-math></inline-formula> is the girth of the graph. The information required for the computation is the node degree and the multiplicity of the nodes on both sides of the partition, as well as the eigenvalues of the adjacency matrix of the graph (graph spectrum). In this paper, the result of Blake and Lin is extended to compute the number of cycles of length <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {g} + \textbf {2}, \ldots, \textbf {2}\boldsymbol {g}-\textbf {2}$ </tex-math></inline-formula>, for bi-regular bipartite graphs, as well as the number of 4-cycles and 6-cycles in irregular and half-regular bipartite graphs, with <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {g} \geq \textbf {4}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$\boldsymbol {g} \geq \textbf {6}$ </tex-math></inline-formula>, respectively.

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