Cospectral graphs, GM-switching and regular rational orthogonal matrices of level p

Abstract

The GM-switching, invented by Godsil and McKay, is a powerful method to construct cospectral graphs. In its simplest version, the method can be roughly described as follows: let A(G) be the adjacency matrix of a graph G on n vertices, let Q=diag(R,In−2m) be a block diagonal matrix, where R=1mJ−I2m is an orthogonal matrix of order 2m, J is the all-one matrix and Ik denotes the identity matrix of order k. If G satisfies certain conditions, then QTA(G)Q is an adjacency matrix of a graph G′ which is cospectral with G (with cospectral complements).In this paper, we consider the problem of replacing the above R with another rational orthogonal matrix U of order 2p (p an odd prime). We characterize all graphs G admitting a GM-switching corresponding to Q=diag(U,In−2p), i.e., such that QTA(G)Q is an adjacency matrix of a graph G′, and hence giving a counterpart of the GM-switching method for generating cospectral graphs (with cospectral complements), which can be used to construct pairs of non-isomorphic cospectral graphs that are not obtainable by the original GM-switching method.Next, we consider the problem of how to determine whether a given graph has a non-trivial GM-switching associated with some U. A simple algebraic condition is presented which partially answers this question.