Nordhaus–Gaddum-type results for the Steiner Wiener index of graphs

Abstract

© 2016 Elsevier B.V. The Wiener index W of a connected graph G with vertex set V(G) is defined as W=∑u,v∈V(G)d(u,v) where d(u,v) stands for the distance between the vertices u and v of G. For S⊆V(G), the Steiner distance d(S) of the vertices of S is the minimum size of a connected subgraph of G whose vertex set contains S. The kth Steiner Wiener index SWk(G) of G is defined as the sum of Steiner distances of all k-element subsets of V(G). In 2005, Zhang and Wu studied the Nordhaus–Gaddum problem for the Wiener index. We now obtain analogous results for SWk, namely sharp upper and lower bounds for SWk(G)+SWk(G¯) and SWk(G)⋅SWk(G¯), valid for any connected graph G whose complement G¯ is also connected.