On spectral invariants of the [formula omitted]-mixed adjacency matrix

Abstract

Let Gˆ be a mixed graph and α∈[0,1]. Let Dˆ(Gˆ) and Aˆ(Gˆ) be the diagonal matrix of vertex degrees and the mixed adjacency matrix of Gˆ, respectively. The α-mixed adjacency matrix of Gˆ is the matrix Aˆα(Gˆ)=αDˆ(Gˆ)+(1−α)Aˆ(Gˆ).We study some properties of Aˆα(Gˆ) associated with some type of mixed graphs, namely quasi-bipartite and pre-bipartite mixed graphs. A spectral characterization for pre-bipartite and some class of quasi-bipartite mixed graphs is given. For a mixed graph Gˆ we exploit the problem of finding the smallest α for which Aˆα(Gˆ) is positive semi-definite. This problem was proposed by Nikiforov in the context of undirected graphs. It is proven here that, for a mixed graph this number is not greater than 12 and that a connected mixed graph Gˆ with n≥2 is quasi-bipartite if and only if this number is exactly 12. The spread of the α-mixed adjacency matrix is the difference among the largest and the smallest α-mixed adjacency eigenvalue. Upper and lower bounds for the spread of the α- mixed adjacency matrix are obtained. The α-mixed Estrada index of Gˆ is the sum of the exponentials of the eigenvalues of Aˆα(Gˆ). In this paper, bounds for the eigenvalues of Aˆα(Gˆ) are established and, using these bounds some sharp bounds on the mixed Estrada index of Aˆα(Gˆ) are presented.