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

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 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${M}$ </tex-math></inline-formula> -polynomials of two nanotubes, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$VC_{5} C_{7} [p,q]$ </tex-math></inline-formula> and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$HC_{5} C_{7} [p,q]$ </tex-math></inline-formula> . By applying calculus on these <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${M}$ </tex-math></inline-formula> -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.