On the Steiner hyper-Wiener index of a graph

TL;DR

This paper introduces the Steiner hyper-Wiener index, explores its relation to the Steiner Hosoya polynomial, and develops efficient computation methods for median and grid graphs, enhancing understanding of graph distances.

Contribution

It defines the Steiner hyper-Wiener index, relates it to the Steiner Hosoya polynomial, and provides a new efficient method for calculating it in median and grid graphs.

Findings

Relation between Steiner hyper-Wiener index and Steiner Hosoya polynomial

Efficient computation method for median graphs

Closed formulas for grid graphs

Abstract

In this paper, we study the Steiner hyper-Wiener index of a graph, which is obtained from the standard hyper-Wiener index by replacing the classical graph distance with the Steiner distance. It is shown how this index is related to the Steiner Hosoya polynomial, which generalizes similar result for the standard hyper-Wiener index. Next, we show how the Steiner $3$-hyper-Wiener index of a modular graph can be expressed by using the classical graph distances. As the main result, a method for computing this index for median graphs is developed. Our method makes computation of the Steiner $3$-hyper-Wiener index much more efficient. Finally, the method is used to obtain the closed formulas for the Steiner $3$-Wiener index and the Steiner $3$-hyper-Wiener index of grid graphs.