On the construction of cospectral nonisomorphic bipartite graphs

TL;DR

This paper develops a new method for constructing bipartite graphs that are cospectral for both adjacency and normalized Laplacian matrices, simplifying previous proofs and extending existing constructions.

Contribution

It introduces a partitioned tensor product approach to construct cospectral bipartite graphs, simplifying proofs and generalizing previous characterizations of isomorphism.

Findings

Constructed bipartite graphs cospectral for adjacency and normalized Laplacian matrices.

Simplified proof of bipartite case of Godsil and McKay's construction.

Extended Ji et al.'s isomorphism characterization to biregular bipartite graphs.

Abstract

In this article, we construct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices using partitioned tensor product. This extends the construction of Ji, Gong, and Wang \cite{ji-gong-wang}. Our proof of the cospectrality of adjacency matrices simplifies the proof of the bipartite case of Godsil and McKay's construction \cite{godsil-mckay-1976}, and shows that the corresponding normalized Laplacian matrices are also cospectral. We partially characterize the isomorphism in Godsil and McKay's construction, and generalize Ji et al.'s characterization of the isomorphism to biregular bipartite graphs. The essential idea in characterizing the isomorphism uses Hammack's cancellation law as opposed to Hall's marriage theorem used by Ji et al.