Exponentially many graphs have a Q-cospectral mate

We develop an algorithm for computing the characteristic polynomial of matrices related to threshold graphs. We use this as tool to exhibit, for any natural number n≥4, 2n−4 graphs with n vertices that have a non isomorphic pair with the same signless Laplacian spectrum. We also show how to construct infinite families of pairs of non isomorphic graphs having the same Q-spectrum.