Colorings and Aα spectral radius of (join) digraphs

Let Gn,r be the set of digraphs of order n with dichromatic number r. Let in-tree be a directed tree with n vertices which the outdegree of each vertex is at most one. In this paper, we obtain the digraph which has the minimal A? spectral radius of the join of in-trees with dichromatic number r. Moreover, we characterize the digraph which has the maximal A? spectral radius in Gn,r by using a new method.