On the A _α spectral radius of digraphs with given parameters

Let G be a digraph and (Formula presented.) be the adjacency matrix of G. Let (Formula presented.) be the diagonal matrix with outdegrees of vertices of G. For any real (Formula presented.), define the matrix (Formula presented.) as (Formula presented.) The largest modulus of the eigenvalues of (Formula presented.) is called the (Formula presented.) spectral radius of G. In this paper, we determine the digraphs which attain the maximum (or minimum) (Formula presented.) spectral radius among all strongly connected digraphs with given parameters such as girth, clique number, vertex connectivity or arc connectivity. We also propose an open problem.