Structure of Trees with Respect to Nodal Vertex Sets

Abstract

Let $T$ be a tree with a given adjacency eigenvalue $\lambda$. In this paper, by using the $\lambda$-minimal trees, we determine the structure of trees with a given multiplicity of the eigenvalue $\lambda$. Furthermore, we consider the relationship between the structure of trees and the eigensystem of a given Laplacian eigenvalue.

14 pages, 11 figures, 3 tables