Abstract
Two graphs G and H are cospectral if they share the same spectrum. Constructing cospectral non-isomorphic graphs has been investigated for many years and various constructions are known in the literature, among which, the GM-switching, invented by Godsil and McKay in 1982, is proved to a simple and powerful one. Motivated by this, we address the following problem: “For which graphs G with adjacency matrix A(G), does there exist a rational orthogonal matrix Q (not a permutation matrix) with constant row sum, such that QTA(G)Q is a (0,1)-matrix?” We focus on a special case that 2Q is an integral matrix with one fully indecomposable block. Some partial answers to the above question are given by associating a directed graph ΓG with G. We find that there exists an interesting relationship between the existence of the above Q and the structure of ΓG. Based on the results obtained, we present a new method for the construction of cospectral graphs.
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