Wiener index and vulnerability parameters of graphs

Let G be a finite simple graph with vertex set V(G). We define the Wiener index of G as W(G)=∑_{u,v}∈V(G)d(u,v), where d(u,v) is the distance between u and v. This topological index was defined by Wiener in 1947 for studying the structural graphs of molecules (Wiener, 1947 [17]) (although, Wiener's definition included a 1/2 multiple in front of the sum). Since that time, many theoretical results have been derived. The results we present concern the graph parameters integrity, toughness, tenacity and binding number. Lower bounds on these parameters have been used extensively to determine properties of graphs. For each of the four parameters, we provide a best possible upper bound on the Wiener index of G that guarantees a specified lower bound on that parameter. This provides a continuation of results provided in Feng et al. (2017).