Lifshitz tails for spectra of Erd.s.Rényi random graphs

We consider the discrete Laplace operator .(N) on Erd.s.Rényi random graphs with N vertices and edge probability p/N. We are interested in the limiting spectral properties of .(N) as N.. in the subcritical regime 0<p<1 where no giant cluster emerges. We prove that in this limit the expectation value of the integrated density of states of .(N) exhibits a Lifshitz-tail behavior at the lower spectral edge E=0