Graphs with at most one Generalized Cospectral Mate

Let G be an n-vertex graph with adjacency matrix A, and W=e, Ae, , A^n-1e be the walk matrix of G, where e is the all-one vector. In Wang J. Combin. Theory, Ser. B, 122 (2017): 438-451, the author showed that any graph G is uniquely determined by its generalized spectrum (DGS) whenever 2^- n/2 W is odd and square-free. In this paper, we introduce a large family of graphsFₙ=\n-vertex graphs~ G\, 2^{- n/2 W =p²b and rankW=n-1 over Z/pZ\}, where b is odd and square-free, p is an odd prime and p b. We prove that any graph in Fₙ either is DGS or has exactly one generalized cospectral mate up to isomorphism. Moreover, we show that the problem of finding the generalized cospectral mate for a graph in Fₙ is equivalent to that of generating an appropriate rational orthogonal matrix from a given integral vector. This equivalence essentially depends on a surprising property of graphs in terms of generalized spectra, which states that any symmetric integral matrix generalized cospectral with the adjacency matrix of some graph must be an adjacency matrix. Based on this equivalence, we develop an efficient algorithm to decide whether a given graph in Fₙ is DGS and further to find the unique generalized cospectral mate when it is not. We give some experimental results on graphs with at most 20 vertices, which suggest that Fₙ may have a positive density (nearly 3\%) and possibly almost all graphs in Fₙ are DGS as n. This gives supporting evidence for Haemers' conjecture that almost all graphs are determined by their spectra.