On a lower bound for the eccentric connectivity index of graphs

The eccentric connectivity index of a graph $G$, denoted by $\xi^{c}(G)$, defined as $\xi^{c}(G)$ = $\sum_{v \in V(G)}\epsilon(v) \cdot d(v)$, where $\epsilon(v)$ and $d(v)$ denotes the eccentricity and degree of a vertex $v$ in a graph $G$, respectively. The volcano graph $V_{n,d}$ is a graph obtained from a path $P_{d+1}$ and a set $S$ of $n-d-1$ vertices, by joining each vertex in $S$ to a central vertex or vertices of $P_{d+1}$. In (A lower bound on the eccentric connectivity index of a graph, Discrete Applied Math., 160, 248 to 258, (2012)), Morgan et al. proved that $\xi^{c}(G) \geq \xi^{c}(V_{n,d})$ for any graph of order $n$ and diameter $d \geq 3$. In this paper, we present a short and simple proof of this result by considering the adjacency of vertices in graphs.