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 $\alpha\in\left[ 0,1\right],$definethematrix$A_{\alpha}\left(G\right) $ as $$A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right)$$ where $0\le \alpha \le 1$. This paper gives several results about the $A_{\alpha }$-matrices of trees. In particular, it is shown that if $T_{\mathrm{\Delta }}$ is a tree of maximal degree $\mathrm{\Delta },$ then the spectral radius of $A_{\alpha }(T_{\mathrm{\Delta }})$ satisfies the tight inequality $$\rho (A_{\alpha }(T_{\mathrm{\Delta }}))<\alpha \mathrm{\Delta }+2(1-\alpha )\sqrt{\mathrm{\Delta }-1}\mathrm{.}$$ 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.