M-Polynomials and Degree-Based Topological Indices of VC5C7[p,q] and HC5C7[p,q] Nanotubes

Jia-Bao Liu,Waqas Nazeer,Muhammad Yousaf,Mustafa Habib,Muhammad Younas

doi:10.1109/access.2019.2907667

Abstract

Analysts are creating materials, for example, a carbon nanotube-based composite created by NASA that bends when a voltage is connected. Applications incorporate the use of an electrical voltage to change the shape (transform) of airship wings and different structures. Topological indices are numerical parameters of a molecular graph which characterize its topology and are usually graph invariant. Topological indices are used, for example, in the development of quantitative structure-activity relationships (QSARs) in which the biological activity or other properties of molecules are correlated with their chemical structure. Topological indices catch symmetry of molecular structures and help us to predict properties, for example, boiling points, viscosity, and the radius of gyrations of nanotubes. In this paper, we compute M-polynomials of two nanotubes, VC 5 C 7 [p, q] and HC 5 C 7 [p, q]. By applying calculus on these M-polynomials, we produce formulas of numerous degree-based topological indices, which are functions relying on parameters of the structure and, in combination, decide properties of the concerned polymeric compound.

Highlights

Carbon nanotubes are allotropes of carbon with a cylindrical nanostructure
The flexibility and strength of carbon nanotubes makes them of potential use in controlling other nanoscale structures, which recommends they will have a critical job in nanotechnology engineering
We focus on the degree-based combinatorial facts of VC5C7[p, q] and HC5C7[p, q] nanotubes

Summary

INTRODUCTION

Carbon nanotubes are allotropes of carbon with a cylindrical nanostructure. These cylindrical carbon particles have surprising properties, which are important for nanotechnology, optics, electronics and different fields of materials science and innovation [1]–[3]. Topological indices remain invariant upto graph isomorphism and help to predict many properties of chemical compounds and nanomaterials, for example, boiling points, viscosity and the radius of gyrations without going to lab [53]–[56]. Another emerging field is Cheminformatics, in which quantitative structure-activity (QSAR) and Structureproperty (QSPR) relationship are used to see the biological activity and properties of nanomaterial. The molecular graphs of carbon nanotubes VC5C7[p, q] and HC5C7[p, q] are shown in Figure 1 and Figure 2 respectively The structures of these nanotubes consist of cycles C5 and C7 (C5C7 net which is a trivalent decoration constructed by alternating C5 and C7) by different compound. We give M-polynomials and by applying fundamental calculus on M polynomials, we compute Zagreb indices, Randić indices, Symmetric division indices, inverse sum indices etc of VC5C7[p, q] and HC5C7[p, q] nanotubes

BASIC DEFINITION AND LITERATURE REVIEW

Augmented Zagreb Index

CONCLUDING REMARKS