# Spectra of eccentricity matrices of graphs

## TL;DR

This paper investigates the spectral properties of eccentricity matrices of graphs, confirming a conjecture about their least eigenvalue for trees and analyzing spectra and inertia for various graph classes.

## Contribution

It affirms a conjecture on the least eigenvalue of eccentricity matrices of trees and explores spectra and inertia of these matrices across different graph classes.

## Key findings

- Confirmed conjecture on least eigenvalue for trees.
- Analyzed spectra of eccentricity matrices for various graphs.
- Studied inertia of eccentricity matrices.

## Abstract

The eccentricity matrix of a connected graph $G$ is obtained from the distance matrix of $G$ by retaining the largest distances in each row and each column, and setting the remaining entries as $0$. In this article, a conjecture about the least eigenvalue of eccentricity matrices of trees, presented in the article [Jianfeng Wang, Mei Lu, Francesco Belardo, Milan Randic. The anti-adjacency matrix of a graph: Eccentricity matrix. Discrete Applied Mathematics, 251: 299-309, 2018.], is solved affirmatively. In addition to this, the spectra and the inertia of eccentricity matrices of various classes of graphs are investigated.