A Method for Computing the Edge-Hyper-Wiener Index of Partial Cubes and an Algorithm for Benzenoid Systems

TL;DR

This paper introduces a method to compute the edge-hyper-Wiener index for partial cubes and benzenoid systems, providing new algorithms and correcting existing formulas for specific molecular graph classes.

Contribution

The paper presents a novel method for calculating the edge-hyper-Wiener index of partial cubes and develops an algorithm for benzenoid systems, improving computational techniques in chemical graph theory.

Findings

Method applicable to partial cubes and trees

Algorithm for benzenoid systems developed

Corrected formulas for linear polyacenes

Abstract

The edge-hyper-Wiener index of a connected graph $G$ is defined as $WW_e(G) = \frac{1}{2}\sum_{\lbrace e,f\rbrace \subseteq E(G)}d(e,f) + \frac{1}{2}\sum_{\lbrace e,f\rbrace \subseteq E(G)}d(e,f)^2$. We develop a method for computing the edge-hyper-Wiener index of partial cubes, which constitute a large class of graphs with a lot of applications. It is also shown how the method can be applied to trees. Furthermore, an algorithm for computing the edge-hyper-Wiener index of benzenoid systems is obtained. Finally, the algorithm is used to correct already known closed formulas for the edge-Wiener index and the edge-hyper-Wiener index of linear polyacenes.