The Wiener Polarity Index of Benzenoid Systems and Nanotubes

Niko Tratnik

doi:10.5562/cca3291

Abstract

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.

Highlights

B ENZENOID systems represent a mathematical model for molecules called benzenoid hydrocarbons and form one of the most important classes of chemical graphs.[1]
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 used to find a closed formula for the Wiener polarity index of any benzenoid system

Summary

INTRODUCTION

B ENZENOID systems ( called hexagonal systems) represent a mathematical model for molecules called benzenoid hydrocarbons and form one of the most important classes of chemical graphs.[1]. Using the terminology of graph theory, a tubulene G is defined to be the finite graph induced by all the hexagons of HTT that lie between c1 and c2, where c1 and c2 are two vertex-disjoint cycles of HTT encircling the axis of the cylinder Any such hexagon (between c1 and c2) is called a hexagon of a tubulene G and the number of these hexagons will be denoted by h(G). Four (or even less), it can happen that two distinct hexagons have two common edges, which is not usual for the considered molecules By including such a requirement, a tubulene can not contain a cycle of length four (or less), which is essentially used in Theorem 3.5

A CUT METHOD FOR BENZENOID SYSTEMS AND TUBULENES

A CLOSED FORMULA FOR BENZENOID SYSTEMS

CONCLUSION