  We study the spectrum of adjacency matrices of random graphs. We develop two
techniques to lower bound the mass of the continuous part of the spectral
measure or the density of states. As an application, we prove that the spectral
measure of bond percolation in the two dimensional lattice contains a
non-trivial continuous part in the supercritical regime. The same result holds
for the limiting spectral measure of a supercritical Erdos-Renyi graph and for
the spectral measure of a unimodular random tree with at least two ends. We
give examples of random graphs with purely continuous spectrum.
