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 ${\lambda }_{\alpha 1},{\lambda }_{\alpha 2},\dots ,{\lambda }_{\alpha n}$ 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 ${\mathcal{G}}_n^m$.

14 pages, 6 figures