Spectra of Edge-Independent Random Graphs

Let G be a random graph on the vertex set {1, 2, . . . , n} such that edges in G are determined by independent random indicator variables, while the probabilities pij for {i, j} being an edge in G are not assumed to be equal. Spectra of the adjacency matrix and the normalized Laplacian matrix of G are recently studied by Oliveira and Chung-Radcliffe. Let A be the adjacency matrix of G, Ā = E(A), and ∆ be the maximum expected degree of G. Oliveira first proved that asymptotically almost surely ‖A − Ā‖ = O( √ ∆ lnn) provided ∆ > C lnn for some constant C. ChungRadcliffe improved the hidden constant in the error term using a new Chernofftype inequality for random matrices. Here we prove that asymptotically almost surely ‖A − Ā‖ 6 (2 + o(1)) √ ∆ with a slightly stronger condition ∆ ln n. For the Laplacian matrix L of G, Oliveira and Chung-Radcliffe proved similar results ‖L − L̄‖ = O( √ lnn/ √ δ) provided the minimum expected degree δ > C ′ lnn for some constant C ′; we also improve their results by removing the √ lnn multiplicative factor from the error term under some mild conditions. Our results naturally apply to the classical Erdős-Rényi random graphs, random graphs with given expected degree sequences, and bond percolation of general graphs.