On the Eccentric Graph of Trees

Abstract

We consider the eccentric graph of a graph G , denoted by e c c ( G ) , which has the same vertex set as G , and two vertices in the eccentric graph are adjacent if and only if 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 e c c ( G ) and the complement of G , and show that the previous necessary condition given in the literature is inadequate. Also, we obtain that the diameter of e c c ( T ) is at most 3 for any tree and get some characterizations of the eccentric graph of trees.