  We study random graphs with arbitrary distributions of expected degree and
derive expressions for the spectra of their adjacency and modularity matrices.
We give a complete prescription for calculating the spectra that is exact in
the limit of large network size and large vertex degrees. We also study the
effect on the spectra of hubs in the network, vertices of unusually high
degree, and show that these produce isolated eigenvalues outside the main
spectral band, akin to impurity states in condensed matter systems, with
accompanying eigenvectors that are strongly localized around the hubs. We also
give numerical results that confirm our analytic expressions.
