Wiener index and Steiner 3-Wiener index of graphs

Abstract

Let S be a set of vertices of a connected graph G. The Steiner distance of S is the minimum size of a connected subgraph of G containing all the vertices of S. The sum of all Steiner distances on sets of size k is called the Steiner k-Wiener index. A graph G is modular if for every three vertices x,y,z there exists a vertex w that lies on the shortest path between every two vertices of x,y,z. The Steiner 3-Wiener index of a modular graph is obtained in terms of its Wiener index. As concrete examples, we discuss the case of Fibonacci, Lucas cubes and the Cartesian product of modular graphs. The Steiner Wiener index of block graphs is also studied. � 2021 World Scientific Publishing Company.