Some bounds for the Aα-spectral radius of a digraph

Abstract

Let [Formula: see text] be a digraph of order [Formula: see text]. Let [Formula: see text] and [Formula: see text] be respectively, the adjacency matrix and the diagonal matrix of vertex out-degrees of [Formula: see text] The generalized adjacency matrix [Formula: see text] of [Formula: see text] is defined as [Formula: see text] The spectral radius of the matrix [Formula: see text] is called the [Formula: see text]-spectral radius or the generalized adjacency spectral radius of [Formula: see text]. In this paper, we obtain some sharp upper and lower bounds for the [Formula: see text]-spectral radius, in terms of the number [Formula: see text] of vertices, the vertex out-degrees, the average [Formula: see text]-out-degrees, the average [Formula: see text]-in-degrees of the vertices of [Formula: see text] and the parameter [Formula: see text]. We characterize the extremal digraphs attaining these bounds. We highlight the importance of our results by means of some examples and conclude that the bounds obtained are incomparable.