An algorithm for the calculation of the hyper-Wiener index of benzenoid hydrocarbons

New Bounds for Topological Indices on Trees through Generalized Methods

Topological indices are useful for predicting the physicochemical behavior of chemical compounds. A main problem in this topic is finding good bounds for the indices, usually when some parameters of the graph are known. The aim of this paper is to use a unified approach in order to obtain several new inequalities for a wide family of topological indices restricted to trees and to characterize the corresponding extremal trees. The main results give upper and lower bounds for a large class of topological indices on trees, fixing or not the maximum degree. This class includes the first variable Zagreb, the Narumi–Katayama, the modified Narumi–Katayama and the Wiener index.

Hosoya polynomial of zigzag polyhex nanotorus

The Hosoya polynomial of a molecular graph G is defined as H(G,?)=?{u,v}V?(G) ?d(u,v), where d(u,v) is the distance between vertices u and v. The first derivative of H(G,?) at ?=1 is equal to the Wiener index of G, defined as W(G)?{u,v}?V(G)d(u,v). The second derivative of 1/2 ?H(G, ?) at ?=1 is equal to the hyper-Wiener index, defined as WW(G)+1/2?{u,v}?V(G)d(u,v)?. Xu et al.1 computed the Hosoya polynomial of zigzag open-ended nanotubes. Also Xu and Zhang2 computed the Hosoya polynomial of armchair open-ended nanotubes. In this paper, a new method was implemented to find the Hosoya polynomial of G = HC6[p,q], the zigzag polyhex nanotori and to calculate the Wiener and hyper Wiener indices of G using H(G,?).

Note of the hyper-Wiener index

The hyper-Wiener index WW of a chemical tree T is defined as the sum of the products n1n2, over all pairs ?,? of vertices of T, where n1 and n2 are the number of vertices of T, lying on the two sides of the path which connects ? and ?. We examine a slight modification WWW of the hyper-Wiener index defined as the sum of the products n1n2n3, over all pairs ?,? of vertices of T, where n3 is the number of vertices of T, lying between ? and ?. It is found that WWW correlates significantly better with various physico-chemical properties of alkanes than WW. Lower and upper bounds for WWW, and an approximate relation between WWW and WW are obtained.

Three methods for calculation of the hyper-Wiener index of molecular graphs

The hyper-Wiener index WW of a graph G is defined as WW(G) = (summation operator d (u, v)(2) + summation operator d (u, v))/2, where d (u, v) denotes the distance between the vertices u and v in the graph G and the summations run over all (unordered) pairs of vertices of G. We consider three different methods for calculating the hyper-Wiener index of molecular graphs: the cut method, the method of Hosoya polynomials, and the interpolation method. Along the way we obtain new closed-form expressions for the WW of linear phenylenes, cyclic phenylenes, poly(azulenes), and several families of periodic hexagonal chains. We also verify some previously known (but not mathematically proved) formulas.

Polynomial expressions for the hyper-Wiener index of extended hydrocarbon networks

General expressions are derived for the hyper-Wiener index (WW) for several series of hydrocarbons, both benzenoid and non-benzenoid, including some two-dimensional networks. Indeed, such expressions were found for all but one series studied. Methods are also described for eliminating the register overflow problem that besets computer-based approaches to calculating WW for large structures by means of the distance matrix.

A linear algorithm for the Hyper-Wiener index of chemical trees

An algorithm with a complexity linear in the number of vertices is proposed for the computation of the Hyper-Wiener index of chemical trees. This complexity is the best possible. Computational experience for alkanes is reported.