The spectra of 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. The Hermitian adjacency matrix of a mixed graph G of order n is the n x n matrix H(G) = (h(ij)), where h(ij) = -h(ij) = i (with i = root-1) if there exists an arc from vi to v(j) (but no arc from v(j) to vi), h(ij)= h(ij) = 1 if there exists an edge (and no arcs) between vi and v(j), and h(ij) = h(ij) = 0 otherwise (if vi and v(j) are neither joined by an edge nor by an arc). Let lambda(1)(G), lambda(2)(G), . . . , lambda(n)(G) be eigenvalues of H(G). The k-th Hermitian spectral moment of G is defined as s(k)(H(G)) = sigma(n)(i=1) lambda(k)(i)(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, <bold> </bold>and as an application, we estimate the Hermitian spectral moments of random mixed graphs. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).