On a lower bound for the eccentric connectivity index of graphs

Abstract

The eccentric connectivity index of a graph G, denoted by (Formula presented), defined as Formula presented), where e(v) and d(v) denotes the eccentricity and degree of a vertex v in a graph G, respectively. The volcano graph Vn,dis a graph obtained from a path Pd+1 and a set S of n-d-1 vertices, by joining each vertex in S to a central vertex/vertices of Pd+1. In [4], Morgan et al. proved that (Formula presented) for any graph of order n and diameter d≥3. In this paper, we present a short and simple proof of this result by considering the adjacency of vertices in graphs.