Sharp bounds of the $A_{\alpha }$-spectral radii of mixed trees

Abstract

A mixed tree is a tree in which both directed arcs and undirected edges may exist. Let $T$ be a mixed tree with $n$ vertices and $m$ arcs, where an undirected edge is counted twice as arcs. Let $A$ be the adjacency matrix of $T$. For $\alpha \in [0,1]$, the matrix $A_{\alpha }$ of $T$ is defined to be $\alpha D^{+}+(1-\alpha )A$, where $D^{+}$ is the the diagonal out-degree matrix of $T$. The $A_{\alpha }$-spectral radius of $T$ is the largest real eigenvalue of $A_{\alpha }$. We will give a sharp upper bound and a sharp lower bound of the $A_{\alpha }$-spectral radius of $T$.