On the Zagreb and Wiener polarity indices of C3-free chemical nanostructures

Abstract

Topological indices are defined to analyze the drug molecular structures and pharmacy characteristics, such as melting point, boiling point, toxicity and other biological activities. The Wiener index W of a graph is the sum of distances between all pairs of vertices of that graph while the Wiener polarity index W_P is the number of pairs vertices such that the distance between them is three. Interesting, these two indices W and W_P are correlated by a linear formula and used to calculate the boiling points of paraffins along with many successful applications in quantitative structure-activity and structure-property relationships investigations. The topological indices based on the degree of vertices play a vital role in chemical characterization and are of great importance in medical drug experiments. Recently an attempt was made to relate W_P with the oldest degree-based indices such as first and second Zagreb indices by imposing triangle-free and quadrangle-free conditions on the graphs. In this present study, we relax the quadrangle-free condition and obtain a new relation as the key result. By applications of main result, we compute W_P for a number of important nanostructures of drug delivery made by alternating squares, hexagons and octagons in a variety of ways and that including all types of designs like sheets, tubes and tori by giving the exact expressions of Zagreb indices for these nanostructures. © 2020 Utilitas Mathematica Publishing Inc.. All rights reserved.