The degree sequences and spectra of scale-free random graphs

We investigate the degree sequences of scale-free random graphs. We obtain a formula for the limiting proportion of vertices with degree d, confirming non-rigorous arguments of Dorogovtsev, Mendes, and Samukhin (Phys Rev Lett 85 (2000), 4633). We also consider a generalization of the model with more randomization, proving similar results. Finally, we use our results on the degree sequence to show that for certain values of parameters localized eigenfunctions of the adjacency matrix can be found. (c) 2005 Wiley Periodicals, Inc.