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 $alphainleft[ 0,1right],$ define the matrix $A_{alpha}left(Gright) $ as [ A_{alpha}left(Gright) =alpha Dleft(Gright) +(1-alpha)Aleft(Gright) ] where $0leqalphaleq1$. 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}))<alphaDelta+2(1-alpha)sqrt{Delta-1}. ] This bound extends previous bounds of Godsil, Lovasz, and Stevanovic. 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.