On the Spectra of General Random Graphs

We consider random graphs such that each edge is determined by an independent random variable, where the probability of each edge is not assumed to be equal. We use a Chernoff inequality for matrices to show that the eigenvalues of the adjacency matrix and the normalized Laplacian of such a random graph can be approximated by those of the weighted expectation graph, with error bounds dependent upon the minimum and maximum expected degrees. In particular, we use these results to bound the spectra of random graphs with given expected degree sequences, including random power law graphs. Moreover, we prove a similar result giving concentration of the spectrum of a matrix martingale on its expectation.