Almost all graphs are determined by their generalized spectrum

Abstract: Haemers' conjecture for the generalized spectrum asserts that almost all graphs are determined by their generalized spectrum, that is, for almost all graphs $G$, every graph with the same spectrum as $G$ whose complement has the same spectrum as the complement of $G$ is isomorphic to $G$. We prove this conjecture by showing that the random graph $G(n,1/2)$ is determined by its generalized spectrum with probability tending to one. The proof combines a local theory of the denominators of the rational orthogonal matrices that relate generalized cospectral graphs, Fourier analysis of random symmetric matrices over finite rings, Seidel switching, inverse Littlewood--Offord theory over finite fields, rank estimates for random symmetric matrices modulo primes, and a study of the rational factors of the characteristic polynomial of a random symmetric $\pm1$ matrix with zero diagonal.