On the Steiner hyper-Wiener index of a graph

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.