Laplacian Estrada index of trees

TL;DR

This paper investigates the Laplacian Estrada index of trees, establishing extremal values for paths and stars, and identifying the second maximal tree, thereby advancing understanding of spectral graph invariants.

Contribution

It proves extremal properties of the Laplacian Estrada index for trees and identifies the second maximal tree, using a novel connection with the Estrada index of line graphs.

Findings

Path $P_n$ has the minimal Laplacian Estrada index among trees.

Star $S_n$ has the maximal Laplacian Estrada index among trees.

The unique tree with the second maximal Laplacian Estrada index is characterized.

Abstract

Let $G$ be a simple graph with $n$ vertices and let $\mu_1 \geqslant \mu_2 \geqslant...\geqslant \mu_{n - 1} \geqslant \mu_n = 0$ be the eigenvalues of its Laplacian matrix. The Laplacian Estrada index of a graph $G$ is defined as $LEE (G) = \sum\limits_{i = 1}^n e^{\mu_i}$. Using the recent connection between Estrada index of a line graph and Laplacian Estrada index, we prove that the path $P_n$ has minimal, while the star $S_n$ has maximal $LEE$ among trees on $n$ vertices. In addition, we find the unique tree with the second maximal Laplacian Estrada index.