# Trees with a large Laplacian eigenvalue multiplicity

## TL;DR

This paper investigates the multiplicities of non-integer Laplacian eigenvalues in trees, providing bounds and characterizing trees with high eigenvalue multiplicities, highlighting the importance of algebraic connectivity.

## Contribution

It offers new bounds on the multiplicities of non-integer Laplacian eigenvalues in trees and characterizes trees with near-maximal multiplicities, focusing on the role of algebraic connectivity.

## Key findings

- Upper bounds on eigenvalue multiplicities are established.
- Trees with high multiplicities are characterized.
- The role of algebraic connectivity in eigenvalue multiplicities is emphasized.

## Abstract

In this paper, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than $1$ are simple and also the multiplicity of Laplacian eigenvalue $1$ has been well studied before. Here we consider the multiplicities of the other (non-integral) Laplacian eigenvalues. We give an upper bound and determine the trees of order $n$ that have a multiplicity that is close to the upper bound $\frac{n-3}{2}$, and emphasize the particular role of the algebraic connectivity.