Let G be a graph with adjacency matrix and let be the diagonal matrix of the degrees of G. For every real , define the matrix as
This paper gives several results about the -matrices of trees. In particular, it is shown that if is a tree of maximal degree Δ, then the spectral radius of satisfies the tight inequality which implies previous bounds of Godsil, Lovász, and Stevanović. The proof is deduced from some new results about the -matrices of Bethe trees and generalized Bethe trees.
In addition, several bounds on the spectral radius of of general graphs are proved, implying tight bounds for paths and Bethe trees.