On the $A_\alpha$ spectral radius and $A_\alpha$ energy of non-strongly connected digraphs

Abstract

Let $A_\alpha(G)$ be the $A_\alpha$-matrix of a digraph $G$ and $\lambda_{\alpha 1}, \lambda_{\alpha 2}, \ldots, \lambda_{\alpha n}$ be the eigenvalues of $A_\alpha(G)$. Let $\rho_\alpha(G)$ be the $A_\alpha$ spectral radius of $G$ and $E_\alpha(G)=\sum_{i=1}^n \lambda_{\alpha i}^2$ be the $A_\alpha$ energy of $G$ by using second spectral moment. Let $\mathcal{G}_n^m$ be the set of non-strongly connected digraphs with order $n$, which contain a unique strong component with order $m$ and some directed trees which are hung on each vertex of the strong component. In this paper, we characterize the digraph which has the maximal $A_\alpha$ spectral radius and the maximal (minimal) $A_\alpha$ energy in $\mathcal{G}_n^m$.