On the Steiner hyper-Wiener index of a graph
نویسنده
چکیده
For a connected graph G and an non-empty set S ⊆ V (G), the Steiner distance dG(S) among the vertices of S is defined as the minimum size among all connected subgraphs whose vertex sets contain S. This concept represents a natural generalization of the concept of classical graph distance. Recently, the Steiner Wiener index of a graph was introduced by replacing the classical graph distance used in the Wiener index with the Steiner distance. Moreover, the Steiner degree distance and the Steiner Hosoya polynomial also appeared in the literature. In this paper, we focus on the Steiner hyper-Wiener index of graphs. It is shown how this index is related to the Steiner Hosoya polynomial, which generalizes similar result for the classical 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.
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