On the eccentric graph of trees
Sezer Sorgun, Esma Elyemani

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 , denoted by , which has the same vertex set as , and two vertices in the eccentric graph are adjacent iff their distance in is equal to the eccentricity of one of them. In this paper, we present a fundamental requirement for the isomorphism between and the complement of , and show that the previous necessary condition given in the literature is inadequate. Also we obtain that diameter of is at most for any tree and get some characterizations of the eccentric graph of trees.
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Taxonomy
TopicsGraph theory and applications · Graphene research and applications · Fiber-reinforced polymer composites
