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

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 $T$.