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Graphs and Combinatorics

, Volume 35, Issue 6, pp 1673–1693 | Cite as

Cospectral Bipartite Graphs with the Same Degree Sequences but with Different Number of Large Cycles

  • Ali DehghanEmail author
  • Amir H. Banihashemi
Original Paper
  • 53 Downloads

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.

Keywords

Cycle multiplicity Bipartite graphs Tanner graphs Graph spectrum Low-density parity-check (LDPC)  codes Bi-regular bipartite graphs Irregular bipartite graphs Half-regular bipartite graphs Girth Degree sequences of a graph Godsil McKay switching 

Notes

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Copyright information

© Springer Japan KK, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Systems and Computer Engineering DepartmentCarleton UniversityOttawaCanada

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