The Spectral Gap of Random Graphs with Given Expected Degrees

We investigate the Laplacian eigenvalues of a random graphG(n,d) with a given expected degree distribution d. The main result is that w.h.p. G(n,d) has a large subgraph core(G(n,d)) such that the spectral gap of the normalized Laplacian of core(G(n,d)) is ≥ 1− c0d̄ min with high probability; here c0 > 0 is a constant, and d̄min signifies the minimum expected degree. The result in particular applies to sparse graphs with d̄min = O(1) as n→∞, and it is of interest in order to extend known spectral heuristics for random regular graphs to graphs with irregular degree distributions, e.g., power laws. The present paper complements the work of Chung, Lu, and Vu [Internet Mathematics 1, 2003].