Eccentric Connectivity Index of Certain Graphs

Abstract

The eccentric connectivity index of the molecular graph $G$ was proposed by Sharma, Goswami, and Madan in 1997 \cite{17}. This index is defined as ${\xi }^c(G)=\sum _{v\in V(G)}\mathrm{deg}(v)ec(v)$, where $\mathrm{deg}(v)$ is the degree of vertex $v$ in $G$ and eccentricity $ec(v)$ is the largest distance between $v$ and any other vertex of $G$. Thus, in this paper, we established the general formulas for the eccentric connectivity index of joining a special graph to its paths and of joining two different graphs by a path. Proofs were also provided.