The graph spectra and spectral moments of random graphs

Abstract

The spectra of some specific classes of random graphs have received considerable interest in the literature. Here, we investigate the spectra for two random graph models: the FDSM model and the G(n,p) model in which every possible edge in a graph with n vertices occurs with probability p. We determine that under some conditions, the k-th spectral moment of the G(n,p) model is in O(nkpk). Moreover, we give results for the expected number of common neighbors (or cooccurrence) and, more generally, the expected number of walks of length ℓ for the fixed degree sequence model.