On the Aα-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 α∈[0,1], define the matrix Aα(G) as Aα(G)=αD(G)+(1−α)A(G). This paper gives several results about the Aα-matrices of trees. In particular, it is shown that if TΔ is a tree of maximal degree Δ, then the spectral radius of Aα(TΔ) satisfies the tight inequality ρ(Aα(TΔ))\u3cαΔ+2(1−α)Δ−1, which implies previous bounds of Godsil, Lovász, and Stevanović. The proof is deduced from some new results about the Aα-matrices of Bethe trees and generalized Bethe trees. In addition, several bounds on the spectral radius of Aα of general graphs are proved, implying tight bounds for paths and Bethe trees