The Edge-Wiener Index and the Edge-Hyper-Wiener Index of Phenylenes

TL;DR

This paper introduces methods to compute the edge-Wiener and edge-hyper-Wiener indices of phenylenes, a class of molecular graphs, providing explicit formulas for linear phenylenes and reducing complex calculations to quotient tree indices.

Contribution

It develops a novel approach using quotient trees to efficiently calculate edge indices of phenylenes, extending previous methods used for benzenoid systems.

Findings

Derived closed formulas for linear phenylenes.

Reduced index calculations to weighted Wiener indices of quotient trees.

Provided a new method for computing edge-hyper-Wiener index of phenylenes.

Abstract

Besides the well known Wiener index, which sums up the distances between all the pairs of vertices, and the hyper-Wiener index, which includes also the squares of distances, the edge versions of both indices attracted a lot of attention in the recent years. In this paper we consider the edge-Wiener index and the edge-hyper-Wiener index of phenylenes, which represent an important class of molecular graphs. For an arbitrary phenylene, four quotient trees based on the elementary cuts are defined in a similar way as it was previously done for benzenoid systems. The computation of the edge-Wiener index of the phenylene is then reduced to the calculation of the weighted Wiener indices of the corresponding quotient trees. Furthermore, a method for computing the edge-hyper-Wiener index of phenylenes is described. Finally, the application of these results gives closed formulas for the edge-Wiener index and the edge-hyper-Wiener index of linear phenylenes.