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

• Published in: IEEE transactions on information theory Vol. 65; no. 6; pp. 3778 - 3789
• Main Authors: Dehghan, Ali, Banihashemi, Amir H.
• Format: Journal Article
• Language: English
• Published: New York IEEE 01-06-2019; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: bi-regular bipartite graphs; Binary system; Bipartite graph; bipartite graphs; Codes; Counting; Counting cycles; cycle multiplicity; Eigenvalues; Eigenvalues and eigenfunctions; Error correcting codes; girth; graph spectrum; Graph theory; Graphs; half-regular bipartite graphs; Indexes; IP networks; irregular bipartite graphs; low-density parity-check (LDPC) codes; Parity check codes; short cycles; Symmetric matrices; Tanner graphs; Vegetation
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2019.2895356

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.