Investigation on Boron Alpha Nanotube by Studying Their M-Polynomial and Topological Indices

Graph theory provides an effective tool such as graph polynomial and topological indices (TIs) to the chemist to analyze the different chemical structures. TIs are the numerical entity deducted from the molecular structure. TI helps to study the relationship between the physicochemical properties and structure of the chemical compound. In this article, we investigate the boron α-nanotube by computing its M-polynomial and then deducing its TI. Results are also shown by plotting the graphs.


Introduction
Chemical graph theory plays an important role in analysis, designing, interpreting, modeling, and understanding chemical substances. e molecular graph is composed of vertices (atoms) and edges (chemical bonds). Chemical graph theory has many applications during the study of chemical substances [1,2]. e chemical graph theory provides different tools for mathematical modeling of molecular structure. is modeling is useful for the analysis of chemical compounds. e analysis of chemical compounds is made conceivable by using topological indices (TIs). A large number of TIs are introduced and applied to study pharmacology and theoretical chemistry [3,4]. e first TI which correlated with the boiling points of alkanes was introduced by H. Wiener called the Wiener index in 1947 [5]. Until now, thousands of indices are designed and used in chemical graph theory [6]. A degreedependent topological index for the graph G is defined as follows: Equation (1) is rewritten by counting the same enddegree edges in chemical graph as follows: where d x , d y � j, k and the total number of edges xy is denoted by m jk . Some important TIs are described in [7]. Reduced reciprocal Randić index [8] is defined as  [11] is defined as EM 1 (G) � xy∈E G (d x + d y − 2) 2 . General sum connectivity e search for small size, low cost, and high efficient materials is the most intersecting topic nowadays. To get this goal, there is a need to study the chemical and physical behavior of chemical substances. So, nanotechnology becomes the most important field in the twenty-first century. By using the chemical graph theory technique, nanostructures are transformed into a mathematical model and then inspected under numerous parameters. Due to attractive features such as work function, transport property, electronic structure, and structural stability, boron α-nanotubes have gained an important place in modern times [18,19]. e boron α-nanotube is constructed by boron α-nanosheet consisting of q column and l rows. ere are three methods to connecting the first and last column of boron α-nanosheet: armchair, zigzag, and chiral [20].
On the basis of rows, there are two types of boron α-nanotube such as l ≡ 0 (mod 3) shown in Figure 1(a) and l ≡ 2 (mod 3) shown in Figure 1(b). In the present study, we consider the armchair connecting of boron α-sheet to form a boron α-nanotube for l ≡ 0 (mod 3) and symbolize as B α NT lq shown in Figure 2. Vertex partition of B α NT lq is presented in Table 2, and edge partition is in Table 3.

Topological index Derivation from
Proof.

Conclusion
We studied the important nanotube known as boron α-nanotube by computing their M-polynomial and then recovered some important degree-based TIs. e graphical representations of the results are presented e results obtained will be useful in helping to solve many problems in the field of chemical analysis.

Data Availability
No data were used to support this research.

Conflicts of Interest
e author(s) declare that there are no conflicts of interest regarding the publication of this paper.