On the Aα spectral radius of digraphs with given parameters

Let G be a digraph and A(G) be the adjacency matrix of G. Let D(G) be the diagonal matrix with outdegrees of vertices of G. For any real alpha is an element of[0,1], define the matrix A alpha(G)as A alpha(G)=alpha D(G)+(1-alpha)A(G). The largest modulus of the eigenvalues of A alpha(G )is called the A alpha spectral radius ofG. In this paper, we determine the digraphs which attain the maximum (or minimum) A alpha 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.