On the $A_\alpha$ spectral radius and $A_\alpha$ energy of digraphs

Let $G$ be a digraph with adjacency matrix $A(G)$ and outdegrees diagonal matrix $D(G)$. For any real $\alpha\in[0,1]$, the $A_\alpha$ matrix $A_\alpha(G)$ of a digraph $G$ is defined as $A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$. The eigenvalue of $A_\alpha(G)$ with the largest modulus is called the $A_\alpha$ spectral radius of $G$. In this paper, we first give some upper bounds for the $A_\alpha$ spectral radius of a digraph and we also characterize the extremal digraphs attaining these bounds. Moreover, we define the $A_\alpha$ energy of a digraph $G$ as $E^{A_\alpha}(G)=\sum\limits_{i=1}^n(\lambda^\alpha_i(G))^2$, where $n$ is the number of vertices and $\lambda^\alpha_i(G)$ $(i=1,2,\ldots,n)$ are the eigenvalues of $A_\alpha(G)$. We obtain a formula for $E^{A_\alpha}(G)$, and give a lower and upper bounds for $E^{A_\alpha}(G)$ and characterize the extremal digraphs that attain the lower and upper bounds. Finally, we characterize the extremal digraphs with maximum and minimum $A_\alpha$ energy among all directed trees and unicyclic digraphs, respectively.