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.