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

Let G ˆ be a mixed graph and α ∈ [ 0 , 1 ]. Let D ˆ (G ˆ) and A ˆ (G ˆ) be the diagonal matrix of vertex degrees and the mixed adjacency matrix of G ˆ , respectively. The α -mixed adjacency matrix of G ˆ is the matrix A ˆ α (G ˆ) = α D ˆ (G ˆ) + (1 − α) A ˆ (G ˆ). We study some properties of A ˆ α (G ˆ) 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 G ˆ we exploit the problem of finding the smallest α for which A ˆ α (G ˆ) 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 1 2 and that a connected mixed graph G ˆ with n ≥ 2 is quasi-bipartite if and only if this number is exactly 1 2. 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 G ˆ is the sum of the exponentials of the eigenvalues of A ˆ α (G ˆ). In this paper, bounds for the eigenvalues of A ˆ α (G ˆ) are established and, using these bounds some sharp bounds on the mixed Estrada index of A ˆ α (G ˆ) are presented. [ABSTRACT FROM AUTHOR]