On the eccentric graph of trees

TL;DR

This paper investigates the properties of eccentric graphs of trees, establishing conditions for their isomorphism with graph complements and characterizing their diameters and structures.

Contribution

It provides a fundamental criterion for isomorphism between eccentric graphs and graph complements, and characterizes the eccentric graphs of trees.

Findings

Diameter of eccentric graph of any tree is at most 3.

Necessary conditions for isomorphism are refined and clarified.

Characterizations of eccentric graphs of trees are provided.

Abstract

We consider the eccentric graph of a graph $G$, denoted by $ecc(G)$, which has the same vertex set as $G$, and two vertices in the eccentric graph are adjacent iff their distance in $G$ is equal to the eccentricity of one of them. In this paper, we present a fundamental requirement for the isomorphism between $ecc(G)$ and the complement of $G$, and show that the previous necessary condition given in the literature is inadequate. Also we obtain that diameter of $ecc(T)$ is at most $3$ for any tree and get some characterizations of the eccentric graph of trees.