Ordering digraphs with maximum outdegrees by their $A_{\alpha }$ spectral radius

Abstract

Let $G$ be a strongly connected digraph with $n$ vertices and $m$ arcs. For any real $\alpha \in [0,1]$, the $A_{\alpha }$ matrix of a digraph $G$ is defined as $$A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G),$$ where $A(G)$ is the adjacency matrix of $G$ and $D(G)$ is the outdegrees diagonal matrix of $G$. The eigenvalue of $A_{\alpha }(G)$ with the largest modulus is called the $A_{\alpha }$ spectral radius of $G$, denoted by ${\lambda }_{\alpha }(G)$. In this paper, we first obtain an upper bound on ${\lambda }_{\alpha }(G)$ for $\alpha \in [\frac{1}{2},1)$. Employing this upper bound, we prove that for two strongly connected digraphs $G_1$ and $G_2$ with $n\ge 4$ vertices and $m$ arcs, and $\alpha \in [\frac{1}{\sqrt{2}},1)$, if the maximum outdegree ${\Delta }^{+}(G_1)\ge 2\alpha (1-\alpha )(m-n+1)+2\alpha$ and ${\Delta }^{+}(G_1)>{\Delta }^{+}(G_2)$, then ${\lambda }_{\alpha }(G_1)>{\lambda }_{\alpha }(G_2)$. Moreover, We also give another upper bound on ${\lambda }_{\alpha }(G)$ for $\alpha \in [\frac{1}{2},1)$. Employing this upper bound, we prove that for two strongly connected digraphs with $m$ arcs, and $\alpha \in [\frac{1}{2},1)$, if the maximum outdegree ${\Delta }^{+}(G_1)>\frac{2m}{3}+1$ and ${\Delta }^{+}(G_1)>{\Delta }^{+}(G_2)$, then ${\lambda }_{\alpha }(G_1)+\frac{1}{4}>{\lambda }_{\alpha }(G_2)$.