Constructing cospectral bipartite graphs

Abstract

Given a bipartite graph G=(V,E) with bipartition V=A∪B, where |A|=|B|. We present a new construction of pairs of cospectral bipartite graphs by “rotating” G in two different ways. More precisely, let l and k be two positive integers with l≠k. Take l copies of A (resp. B) and k copies of B (resp. A), say A1,A2,…,Al and B1,B2,…,Bk (resp. B1′,B2′,…,Bl′ and A1′,A2′,…,Ak′). Let Γ (resp. Γ′) be the graph with vertex set V(Γ)=A1∪A2∪⋯∪Al∪B1∪B2∪⋯∪Bk (resp. V(Γ′)=B1′∪B2′∪⋯∪Bl′∪A1′∪A2′∪⋯∪Ak′), such that the vertices of Γ (resp. Γ′) are adjacent if and only if they are adjacent in G. We show that Γ and Γ′ are cospectral with respect to the adjacency matrix, as well as the normalized Laplacian matrix. Moreover, we give a simple necessary and sufficient condition for Γ and Γ′ to be non-isomorphic. The main ingredient of the proof involves a use of Hall’s Marriage Theorem.