Structure of Trees with Respect to Nodal Vertex Sets

TL;DR

This paper investigates the structural properties of trees related to specific adjacency and Laplacian eigenvalues, identifying how eigenvalue multiplicities influence tree structure using $\lambda$-minimal trees.

Contribution

It introduces the concept of $\lambda$-minimal trees to characterize tree structures with given eigenvalue multiplicities and explores their relation to Laplacian eigensystems.

Findings

Characterization of trees with specified adjacency eigenvalue multiplicities

Identification of $\lambda$-minimal trees as structural archetypes

Analysis of the relationship between tree structure and Laplacian eigensystems

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.