Extremal problems for the eccentricity matrices of complements of trees

TL;DR

This paper investigates extremal properties of the eccentricity matrices of complements of trees, identifying graphs with maximum and minimum spectral radii and energies, and establishing symmetry and bounds related to these matrices.

Contribution

It characterizes extremal graphs for the eccentricity matrices of complements of trees, including spectral radius, eigenvalues, and energy, and provides Nordhaus-Gaddum bounds.

Findings

Identifies the unique tree with minimum and maximum $\\mathcal{E}$-spectral radius in its complement.

Proves symmetry of $\\mathcal{E}$-eigenvalues about zero for complements of trees.

Establishes bounds for the second largest $\\mathcal{E}$-eigenvalue and $\\mathcal{E}$-energy, with extremal graph characterizations.

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 $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$, in which the largest one is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and characterize the extremal graphs. First, we determine the unique tree whose complement has minimum (respectively, maximum) $\mathcal{E}$-spectral radius among the complements of trees. Then, we prove that the $\mathcal{E}$-eigenvalues of the complement of a tree are symmetric about the origin. As a consequence of these results, we characterize the trees whose complement has minimum (respectively, maximum) least $\mathcal{E}$-eigenvalues among the complements of trees. Finally, we discuss the extremal problems for the second largest $\mathcal{E}$-eigenvalue and the $\mathcal{E}$-energy of complements of trees and characterize the extremal graphs. As an application, we obtain a Nordhaus-Gaddum type lower bounds for the second largest $\mathcal{E}$-eigenvalue and $\mathcal{E}$-energy of a tree and its complement.