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×n matrix H(G)=(h_ij), where h_ij=−h_ji=i (with i=−1) if there exists an arc from v_i to v_j (but no arc from v_j to v_i), h_ij=h_ji=1 if there exists an edge (and no arcs) between v_i and v_j, and h_ij=h_ji=0 otherwise (if v_i and v_j are neither joined by an edge nor by an arc). Let λ₁(G),λ₂(G),…,λ_n(G) be eigenvalues of H(G). The k-th Hermitian spectral moment of G is defined as s_k(H(G))=∑_i=1ⁿλ_i^k(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.