Colorings and $A_\alpha$ spectral radius of (join) digraphs

Abstract

Let $\mathcal{G}_{n,r}$ 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 $A_\alpha$ spectral radius of the join of in-trees with dichromatic number $r$. Moreover, we characterize the digraph which has the maximal $A_\alpha$ spectral radius in $\mathcal{G}_{n,r}$ by using a new method.