Abstract

The eccentricity e(u) of a vertex u is the maximum distance of u to any other vertex of G. A vertex v is an eccentric vertex of vertex u if the distance from u to v is equal to e(u). The eccentric digraph ED(G) of a graph (digraph) G is the digraph that has the same vertex as G and an arc from u to v exists in ED(G) if and only if v is an eccentric vertex of u in G. In this paper, we have considered an open problem. Partly we have characterized graphs with specified maximum degree such that ED(G) = G.

Highlights

  • A directed graph or digraph G consists of a finite nonempty set V G called vertex set with vertices and edge set E G of ordered pairs of vertices called arcs; that is E G represents a binary relation on V G .Throughout this paper, a graph is a symmetric digraph; G1,G2,Gk and adding in additional edges from each vertex of Gi to each vertex in Gi 1, for 1 i k 1

  • Buckley [4] defines the eccentric digraph ED G of a graph G as having the same vertex set as G and there is an arc from u to v if v is an eccentric vertex of u

  • We list some results which are quite evident for eccentric digraphs of graphs

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Summary

Introduction

A directed graph or digraph G consists of a finite nonempty set V G called vertex set with vertices and edge set E G of ordered pairs of vertices called arcs; that is E G represents a binary relation on V G .Throughout this paper, a graph is a symmetric digraph; G1,G2 , ,Gk and adding in additional edges from each vertex of Gi to each vertex in Gi 1 , for 1 i k 1. For example consider a symmetric cycle having a pendant vertex adjacent to one of the vertices on the eccentric digraph of a graph G is equal to its complement, ED(G) = G, if and only if G is either a self-centered graph of radius two or G is the union of k 2 complete graphs.

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