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

Abstract

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 \[ A_α\left(G\right) =αD\left(G\right) +(1-α)A\left(G\right) \] 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 $T_{\Delta }$ is a tree of maximal degree $\Delta ,$ then the spectral radius of $A_{\alpha }(T_{\Delta })$ satisfies the tight inequality \[ ρ(A_α(T_Δ))<αΔ+2(1-α)\sqrt{Δ-1}. \] This bound extends previous bounds of Godsil, Lovász, and Stevanović. 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.

19 pages