On the $A_{\alpha}$-spectra of trees

Let $G$ be a graph with adjacency matrix $A(G)$ and let $D(G)$ be the diagonal matrix of the degrees of $G$. For every real $\alpha\in\left[ 0,1\right],$ define the matrix $A_{\alpha}\left(G\right)$ as $$A_{\alpha}\left(G\right) =\alpha D\left(G\right) +(1-\alpha)A\left(G\right)$$ where $0\leq\alpha\leq1$. This paper gives several results about the $A_{\alpha}$-matrices of trees. In particular, it is shown that if $T_{\Delta}$ is a tree of maximal degree $\Delta,$ then the spectral radius of $A_{\alpha}(T_{\Delta})$ satisfies the tight inequality $$\rho(A_{\alpha}(T_{\Delta}))<\alpha\Delta+2(1-\alpha)\sqrt{\Delta-1}.$$ This bound extends previous bounds of Godsil, Lov\'asz, and Stevanovi\'c. The proof is based on some new results about the $A_{\alpha}$-matrices of Bethe trees and generalized Bethe trees. In addition, several bounds on the spectral radius of $A_{\alpha}$ of general graphs are proved, implying tight bounds for paths and Bethe trees.