Leap eccentric connectivity index of some graph operations with subdivided edges

TL;DR

This paper investigates the leap eccentric connectivity index of various graph operations involving subdivision edges, correcting previous assumptions and establishing bounds and formulas for different graph classes.

Contribution

It provides new bounds and explicit formulas for the leap eccentric connectivity index of graphs with subdivided edges, including join and corona graphs, correcting prior misconceptions.

Findings

Counterexample for the subdivision graph eccentricity relation.

Bounds for the leap eccentric connectivity index of subdivided graphs.

Formulas for join and corona graph variants.

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.