On a lower bound for the eccentric connectivity index of graphs

TL;DR

This paper provides a simple proof for a known lower bound on the eccentric connectivity index of graphs, relating it to the volcano graph, and clarifies the conditions under which the bound holds.

Contribution

It offers a concise and straightforward proof of an existing lower bound on the eccentric connectivity index using adjacency considerations.

Findings

The lower bound on the eccentric connectivity index is valid for graphs of order n and diameter d ≥ 3.

The volcano graph minimizes the eccentric connectivity index among such graphs.

A simplified proof technique based on adjacency is presented.

Abstract

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.