# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1907.12062