Minimizers for the energy of eccentricity matrices of trees

TL;DR

This paper investigates the eccentricity matrix of trees, identifying the trees with minimal second largest eigenvalue and minimal energy, thereby advancing understanding of spectral properties related to eccentricity.

Contribution

It characterizes the trees with the smallest second largest eigenvalue and the minimal eccentricity energy, excluding the star, among all trees on n vertices.

Findings

Identifies the unique tree with the minimum second largest $\\mathcal{E}$-eigenvalue (excluding the star).

Characterizes the trees with the minimum $\mathcal{E}$-energy among all trees on $n$ vertices.

Abstract

The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column and leaving zeros in the remaining ones. The eigenvalues of $\mathcal{E}(G)$ are the $\mathcal{E}$-eigenvalues of $G$. The eccentricity energy (or the $\mathcal{E}$-energy) of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we determine the unique tree with the minimum second largest $\mathcal{E}$-eigenvalue among all trees on $n$ vertices other than the star. Also, we characterize the trees with minimum $\mathcal{E}$-energy among all trees on $n$ vertices.