On Eccentric Connectivity Index of Eccentric Graph of Regular Dendrimer

Abstract

The eccentric connectivity index $\xi ^c(G)$ of a connected graph G is defined as $\xi ^c(G) =\sum _{v \in V(G)}{deg(v) e(v)},$ where deg( v) is the degree of vertex v and e( v) is the eccentricity of v. The eccentric graph, $G_e$, of a graph G has the same set of vertices as G; with two vertices u; v adjacent in Ge if and only if either u is an eccentric vertex of v or v is an eccentric vertex of u: In this paper, we obtain a formula for the eccentric connectivity index of the eccentric graph of a regular dendrimer. We also derive a formula for the eccentric connectivity index for the second iteration of eccentric graph of regular dendrimer