The spectra of random mixed graphs

Abstract

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. The Hermitian adjacency matrix of a mixed graph G of order n is the n×n matrix H(G)=(hij), where hij=−hji=i (with i=−1) if there exists an arc from vi to vj (but no arc from vj to vi), hij=hji=1 if there exists an edge (and no arcs) between vi and vj, and hij=hji=0 otherwise (if vi and vj are neither joined by an edge nor by an arc). Let λ1(G),λ2(G),…,λn(G) be eigenvalues of H(G). The k-th Hermitian spectral moment of G is defined as sk(H(G))=∑i=1nλik(G), where k≥0 is an integer. In this paper, we deal with the asymptotic behavior of the spectrum of the Hermitian adjacency matrix of random mixed graphs. We will present and prove a separation result between the largest and remaining eigenvalues of the Hermitian adjacency matrix, and as an application, we estimate the Hermitian spectral moments of random mixed graphs.