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

TL;DR

This paper investigates the spectral properties of the $A_{\alpha}$-matrix for trees, establishing bounds on the spectral radius that extend previous results and apply to various graph classes.

Contribution

It provides new bounds on the spectral radius of $A_{\alpha}$-matrices for trees, especially maximal degree trees, and extends these bounds to general graphs.

Findings

Spectral radius of $A_{\alpha}$ for maximal degree trees is bounded by a specific inequality.

New results on $A_{\alpha}$-matrices of Bethe and generalized Bethe trees.

Bounds on spectral radius for paths, Bethe trees, and general graphs.

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 \[ A_{\alpha}\left(G\right) =\alpha D\left(G\right) +(1-\alpha)A\left(G\right) \] 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.