On Computing the Multiplicity of Short Cycles in Bipartite Graphs Using the Degree Distribution and the Spectrum of the Graph
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Counting short cycles in bipartite graphs is a fundamental problem of interest in the analysis and design of low-density parity-check (LDPC) 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 g in a bi-regular bipartite graph, where g 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, we extend this result in a number of directions. First, we derive a similar result to compute the number of cycles of length g+2, ..., 2g-2, for bi-regular bipartite graphs. Second, using counter-examples, we demonstrate that the information of the degree distribution and the spectrum of a bi-regular bipartite graph is, in general, insufficient to count the cycles of length i ≥ 2g. Third, we consider irregular bipartite graphs, and show that, in general, the information of degree distribution and spectrum alone is not enough to determine the number of cycles of length i (i-cycles), for any i > g, regardless of the value of g. We demonstrate that the same negative result also holds true for half-regular bipartite graphs, and is also applicable to counting g-cycles for g ≥ 6 and g ≥ 8, in irregular and half-regular graphs, respectively. As positive results, we compute the number of 4-cycles and 6-cycles in irregular and half-regular bipartite graphs, with g ≥ 4 and g ≥ 6, respectively, using only the degree distribution and the spectrum of the graph.
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