Abstract
Finding the multiplicity of 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. Recently, Blake and Lin computed the number of shortest cycles (g-cycles, where g is the girth of the graph) in a bi-regular bipartite graph, in terms of the degree sequences and the spectrum (eigenvalues of the adjacency matrix) of the graph (Blake and Lin in IEEE Trans Inf Theory 64(10): 6526–6535, 2018). This result was subsequently extended in Dehghan and Banihashemi (IEEE Trans Inf Theory 65(6):3778–3789, 2019) to cycles of length \(g+2, \ldots , 2g-2\), in bi-regular bipartite graphs, as well as 4-cycles and 6-cycles in irregular and half-regular bipartite graphs, with \(g \ge 4\) and \(g \ge 6\), respectively. In this paper, we complement these positive results with negative results demonstrating that the information of the degree sequences and the spectrum of a bipartite graph is, in general, insufficient to count (a) the i-cycles, \(i \ge 2g\), in bi-regular graphs, (b) the i-cycles for any \(i > g\), regardless of the value of g, and g-cycles for \(g \ge 6\), in irregular graphs, and (c) the i-cycles for any \(i > g\), regardless of the value of g, and g-cycles for \(g \ge 8\), in half-regular graphs. To obtain these results, we construct counter-examples using the Godsil–McKay switching.
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Notes
The cycles are counted using a Matlab program by Jeff Howbert [14]. This program counts all cycles in a simple undirected graph up to a specified size limit, using a backtracking algorithm.
Note that if t is selected to be an even number, the graphs \(G_1\) and \(G_2\) will not be half-regular.
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This paper was presented in part at ISTC 2018, Hong Kong. This research was supported by NSERC Discovery Grant 217239-2013-RGPIN.
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Dehghan, A., Banihashemi, A.H. Cospectral Bipartite Graphs with the Same Degree Sequences but with Different Number of Large Cycles. Graphs and Combinatorics 35, 1673–1693 (2019). https://doi.org/10.1007/s00373-019-02110-6
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DOI: https://doi.org/10.1007/s00373-019-02110-6