# Most Laplacian eigenvalues of a tree are small

## TL;DR

This paper proves that in any tree with n vertices, at most half of the Laplacian eigenvalues exceed the average degree, revealing a fundamental spectral property of trees.

## Contribution

The paper establishes a new upper bound on the number of Laplacian eigenvalues greater than the average degree in trees, advancing spectral graph theory.

## Key findings

- At most loor(n/2) Laplacian eigenvalues are greater than the average degree in an n-vertex tree.
- Provides a spectral bound related to the structure of trees.
- Enhances understanding of eigenvalue distribution in graph Laplacians.

## Abstract

We show that the number of Laplacian eigenvalues greater than the average degree of a tree having $n$ vertices is at most $\lfloor\frac{n}{2} \rfloor$.