Steiner wiener index and connectivity of graphs

Abstract

The Wiener index W(G) of a connected G, introduced by Wiener in 1947, is defined as W(G) = σu,vϵV(G) dg(u,v) where dg(u,v) is the distance between vertices u and v of G. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. Recently, Li et al. introduced the concept of Steiner A:-Wiener index, which is a generalization of classical Wiener index. For a connected graph G of order at least 2 and S ⊆ V(G), the Steiner distance dc(S) of the vertices of S is the minimum size of a connected subgraph whose vertex set contains S. The Steiner k-Wiener index SWk{G) of G is defined by SWk(G) = σs ⊆ v(G), |s|= k dG(S) In 2006 Gutman and Zhang investigated the relation between connectivity and Wiener index of graphs. The graphs with a given number n of vertices and given (vertex or edge) connectivity l, having minimum Steiner Wiener k-index are determined in this paper.