Characterizing Trees with Large Laplacian Energy

TL;DR

This paper explores the ordering of trees based on Laplacian energy, identifying a large class of trees with maximal energy and establishing new bounds on Laplacian eigenvalues.

Contribution

It introduces a new upper bound on the sum of the largest Laplacian eigenvalues for trees with diameter at least four, aiding in characterizing trees with large Laplacian energy.

Findings

Identifies a class of approximately √n trees with the largest Laplacian energy.

Establishes a new upper bound on the sum of the largest Laplacian eigenvalues for certain trees.

Provides insights into the spectral properties of trees with large Laplacian energy.

Abstract

We investigate the problem of ordering trees according to their Laplacian energy. More precisely, given a positive integer $n$, we find a class of cardinality approximately $\sqrt{n}$ whose elements are the $n$-vertex trees with largest Laplacian energy. The main tool for establishing this result is a new upper bound on the sum $S_k(T)$ of the $k$ largest Laplacian eigenvalues of an $n$-vertex tree $T$ with diameter at least four, where $k \in \{1,...,n\}$.