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Finding the multiplicity of cycles in bipartite graphs is a fundamentalproblem of interest in many fields including the analysis and design oflow-density parity-check (LDPC) codes. Recently, Blake and Lin computed thenumber of shortest cycles ($g$-cycles, where $g$ is the girth of the graph) ina bi-regular bipartite graph, in terms of the degree sequences and the spectrum(eigenvalues of the adjacency matrix) of the graph [{\em IEEE Trans. Inform.Theory 64(10):6526--6535, 2018}]. This result was subsequently extended in[{\em IEEE Trans. Inform. Theory, accepted for publication, Dec. 2018}] tocycles of length $g+2, \ldots, 2g-2$, in bi-regular bipartite graphs, as wellas $4$-cycles and $6$-cycles in irregular and half-regular bipartite graphs,with $g \geq 4$ and $g \geq 6$, respectively. In this paper, we complementthese positive results with negative results demonstrating that the informationof the degree sequences and the spectrum of a bipartite graph is, in general,insufficient to count (a) the $i$-cycles, $i \geq 2g$, in bi-regular graphs,(b) the $i$-cycles for any $i > g$, regardless of the value of $g$, and$g$-cycles for $g \geq 6$, in irregular graphs, and (c) the $i$-cycles for any$i > g$, regardless of the value of $g$, and $g$-cycles for $g \geq 8$, inhalf-regular graphs. To obtain these results, we construct counter-examplesusing the Godsil-McKay switching.

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