Spectral Analysis of Normalized Hermitian Laplacian Matrices in Random Mixed Graphs

A mixed graph is a graph that can be obtained from a simple undirected graph by replacing some of the edges by arcs in precisely one of the two possible directions. Let p=p(n) be a function of n such that 0<p<1. Let G^n(p) be a random mixed graph with n vertices in which all arcs are chosen independently with probability p (and an edge is regarded as two oppositely oriented arcs joining the same pair of vertices). In this paper, we study the spectral properties of the normalized Hermitian Laplacian matrices of the random mixed graphs G^n(p) for large n. We characterize the limiting spectral distribution of the normalized Hermitian Laplacian matrices of random mixed graphs. In fact, under the case that p∈(0,1) and np/ln4n→∞, we prove that the empirical distribution of the eigenvalues of the normalized Hermitian Laplacian matrix converges to the semicircle law.