A Study on Radial Graphs

Abstract

In a graph \(G\), the distance \(d(u,v)\) between a pair of vertices \(u\) and \(v\) is the length of a shortest path joining them. The eccentricity \(e(u)\) of a vertex \(u\) is the distance to a vertex farthest from \(u\). The minimum eccentricity is called the radius of the graph and the maximum eccentricity is called the diameter of the graph. The radial graph \(R(G)\) based on \(G\) has the vertex set as in \(G\). Two vertices \(u\) and \(v\) are adjacent in \(R(G)\) if the distance between them in \(G\) is equal to the radius of \(G\). If \(G\) is disconnected, then two vertices are adjacent in \(R(G)\) if they belong to different components. The main objective of this paper is to find a necessary and sufficient condition for a graph to be a radial graph.