Characteristic vertices of weighted trees via perron values

TLDR: In this article, a weighted tree T with algebraic connectivity and a characteristic vertex v and its associated eigenvectors can be described in terms of the Perron value and vector of a nonnegative matrix which can be computed from the branches of T at v, and the machinery of Perron-Frobenius theory can then be used to characterize Type I and Type II trees.

Abstract: We consider a weighted tree T with algebraic connectivity μ, and characteristic vertex v. We show that μ and its associated eigenvectors can be described in terms of the Perron value and vector of a nonnegative matrix which can be computed from the branches of T at v. The machinery of Perron-Frobenius theory can then be used to characterize Type I and Type II trees in terms of these Perron values, and to show that if we construct a weighted tree by taking two weighted trees and identifying a vertex of one with a vertex of the other, then any characteristic vertex of the new tree lies on the path joining the characteristic vertices of the two old trees.