The Wiener polarity index of benzenoid systems and nanotubes

In this paper, we consider a molecular descriptor called the Wiener polarity index, which is defined as the number of unordered pairs of vertices at distance three in a graph. Molecular descriptors play a fundamental role in chemistry, materials engineering, and in drug design since they can be correlated with a large number of physico-chemical properties of molecules. As the main result, we develop a method for computing the Wiener polarity index for two basic and most commonly studied families of molecular graphs, benzenoid systems and carbon nanotubes. The obtained method is then used to find a closed formula for the Wiener polarity index of any benzenoid system. Moreover, we also compute this index for zig-zag and armchair nanotubes

Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

Building upon the notion of the Gutman index SGut(G), Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index SGutk (G) of G is defined by SGutk (G) = ∑S⊆V(G),|S|=k (∏v∈S degG (v)) dG (S), in which dG (S) is the Steiner distance of S and degG (v) is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on SGutk, and then investigate the Nordhaus-Gaddum-type results for the parameter SGutk . We obtain sharp upper and lower bounds of SGutk (G) + SGutk (G) and SGutk (G) · SGutk (G) for a connected graph G of order n, m edges, maximum degree ∆ and minimum degree δ