The Spectrum Estimates for Random Graphs with Given Expected Degrees

We consider the random graph model G(w) for a given expected degree sequence w = (w1, w2,…, wn), where the probability pij of an edge between vertices vi and vj is defined as wiwjρ with ρ = 1/∑_i=1^n w_i . In this paper, we apply a matrix concentration inequality to derive an upper bound on the largest eigenvalue of the adjacency matrix of the random graph with given expected degrees. We further analyze the expectation of the largest eigenvalue of the adjacency matrix and establish its concentration properties. Additionally, by utilizing an extension of the matrix Chernoff inequality that incorporates an intrinsic dimension parameter, we investigate the expectation and tail behavior of the largest eigenvalue of the Laplacian matrix for the random graph model G(w).