On a conjecture involving Laplacian eigenvalues of trees

TL;DR

This paper discusses algorithms for analyzing Laplacian eigenvalues of trees and proposes a conjecture that relates the distribution of these eigenvalues to the average degree of the tree.

Contribution

It introduces efficient algorithms for computing and analyzing Laplacian eigenvalues of trees and formulates a new conjecture about their distribution relative to the average degree.

Findings

Algorithms for eigenvalue counting in trees

A conjecture relating eigenvalues to average degree

Potential for further theoretical validation

Abstract

Motivated by classic tree algorithms, in 1995 we designed a bottom-up $O(n)$ algorithm to compute the determinant of a tree's adjacency matrix $A$. In 2010 an $O(n)$ algorithm was found for constructing a diagonal matrix congruent to $A + xI_n$, $x \in \mathbb{R}$, enabling one to easily count the number of eigenvalues in any interval. A variation of the algorithm allows Laplacian eigenvalues in trees to be counted. We conjecture that for any tree $T$ of order $n \geq 2$, at least half of its Laplacian eigenvalues are less than $\bar{d} = 2 - \frac{2}{n}$, its average vertex degree.