Leap eccentric connectivity index of some graph operations with subdivided edges

Abstract

The leap eccentric connectivity index of $G$ is defined as $$L\xi^{C}(G)=\sum_{v\in V(G)}d_{2}(v|G)e(v|G)$$ where $d_{2}(v|G) $ be the second degree of the vertex $v$ and $e(v|G)$ be the eccentricity of the vertex $v$ in $G$. In this paper, we first give a counterexample for that if $G$ be a graph and $S(G)$ be its the subdivision graph, then each vertex $v\in V(G)$, $e(v|S(G))=2e(v|G)$ by Yarahmadi in \cite{yar14} in Theorem 3.1. And we describe the upper and lower bounds of the leap eccentric connectivity index of four graphs based on subdivision edges, and then give the expressions of the leap eccentric connectivity index of join graph based on subdivision, finally, give the bounds of the leap eccentric connectivity index of four variants of the corona graph.