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

Authors: Deeba Afzal; Farkhanda Afzal; Sabir Hussain; Faryal Chaudhry; Dhana Kumari Thapa
Journal: Journal of Mathematics
 (2022)

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## Abstract

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 boronα-nanotube by computing its M-polynomial and then deducing its TI. Results are also shown by plotting the graphs.


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## Body

## 1. Introduction

Chemical graph theory plays an important role in analysis, designing, interpreting, modeling, and understanding chemical substances. The 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]. The chemical graph theory provides different tools for mathematical modeling of molecular structure. This modeling is useful for the analysis of chemical compounds. The 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].The 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 degree-dependent topological index for the graph G is defined as follows:(1)IG=∑e=xy∈EGfdx,dy.Equation (1) is rewritten by counting the same end-degree edges in chemical graph as follows:(2)IG=∑j≤kmjkfj,k,where dx,dy=j,k and the total number of edges xy is denoted by mjk. Some important TIs are described in [7]. Reduced reciprocal Randić index [8] is defined as RRRG=∑xy∈EGdx−1dy−1. The first arithmetic-geometric index [9] is defined as AG1G=∑xy∈EGdx+dy/2dx⋅dy. SK indices [10] are defined as SKG=∑xy∈EGdx+dy/2, SK1G=∑xy∈EGdx⋅dy/2, and SK2G=∑xy∈EGdx+dy/22. The first Zagreb index in terms of edge degree [11] is defined as EM1G=∑xy∈EGdx+dy−22. General sum connectivity index [12] is described as SCIG=∑xy∈EG1/dx+dy,SCIλG=∑xy∈EGdx+dyλ. Redefined third Zagreb index [13] is commuted as ReZG3G=∑xy∈EGdxdydx+dy.Graph polynomials are the best way to represent a chemical graph in chemical graph theory. In this paper, we study the M-polynomial to analyze the chemical structure. M-polynomial gives the TIs by applying some derivatives and integration operations [7, 14]. In 2015, M-polynomial is introduced in [15]. TIs via M-polynomial is extensively calculated by many researchers nowadays [14, 16, 17]. M-polynomial for the graph G is defined as follows:(3)MGu,v=∑ψ≤j≤k≤Ψmjkujvk.Here,ψ=mindx|x∈VG and Ψ=maxdx|x∈VG. Closed form via M-polynomial of TI is mentioned in Table 1.Table 1
Topological indices derived fromMGu,v.
Topological indexDerivation fromMGu,vRRRG=Du1/2Dv1/2Qu−1Qv−1MGu,v|u=v=1AG1G=1/2DuJSu1/2Sv1/2MGu,v|u=1SKG=1/2Du+DvMGu,v|u=v=1SK1G=1/2DuDvMGu,v|u=v=1Sk2G=1/4Du2JMGu,v|u=1EM1G=Du2Qu−2JMGu,v|u=1SCIG=Su1/2JMGu,v|u=1SCIλG=DuλJMGu,v|u=1ReZG3G=DuDvDu+DvMGu,v|u=v=1The operators used are defined as follows:(4)Dv1/2MGu,v=v∂∂vMGu,v⋅MGu,v,DvMGu,v=v∂∂vMGu,v,Du1/2MGu,v=u∂∂uMGu,v⋅MGu,v,DuMGu,v=u∂∂uMGu,v,Sv1/2MGu,v=∫0vMGu,ttdt⋅MGu,v,JMGu,v=MGu,u,Su1/2MGu,v=∫0uMGt,vtdt⋅MGu,v,QuαMGu,v=uαMGu,v.


## 2. Chemical Graph of Boronα-Nanotube

The 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, boronα-nanotubes have gained an important place in modern times [18, 19]. The boron α-nanotube is constructed by boron α-nanosheet consisting of q column and l rows. There 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 boronα-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 BαNTlq shown in Figure 2. Vertex partition of BαNTlq is presented in Table 2, and edge partition is in Table 3.Figure 1
Boronα-nanosheet BαNSlq: (a) l=0 (mod 3); (b) l=2 (mod 3).
(a)(b)Figure 2
Boronα-nanotube (BαNTlq).Table 2
Vertex partition of boronα-nanotube (BαNTlq).
dx3456Total verticesNumber of verticesq3/2qql−21/6q2l−34/3qlTable 3
Edge partition of boronα-nanotube (BαNTlq).
dx,dyNumber of edges3,31/2q3,5q3,6q4,43/2q4,52q4,6q5,51/2q3l−85,6q2l−5Total edges1/2q7l−4


## 3. M-Polynomial and Topological Indices of Boronα-Nanotube

Theorem 1.
IfBαNTlq represents a boron α-nanotube, then M-polynomial of BαNTlq is MBαNTlqu,v=q/2u3v3+qu3v5+qu3v6+3q/2u4v4+2qu4v5+qu4v6+q/23l−8u5v5+q2l−8u5v6 [21, 22].Theorem 2.
LetBαNTlq represent the boron α-nanotube and MBαNTlqu,v=q/2u3v3+qu3v5+qu3v6+3q/2u4v4+2qu4v5+qu4v6+q/23l−8u5v5+q2l−8u5v6. Then,(1)
RRRBαNTlq=6+45ql−1/221−42−83+205−210−215q(2)
AG1BαNTlq=1/3045+1130ql+1/60452+545+256+1612−5530−120q(3)
SKBαNTlq=37/2ql−35/2q(4)
SK1BαNTlq=195/4ql−249/4q(5)
SK2BαNTlq=98ql−121q(6)
EM1BαNTlq=258ql−352q(7)
SCIBαNTlq=1/2203310+4011ql+1/13201320+8252+1106−39610−60011q(8)
SCIλBαNTlq=1/23⋅10λ+4⋅11λql+1/26λ+5⋅8λ+6⋅9λ−6⋅10λ−10⋅11λq(9)
ReZG3BαNTlq=1035ql−1549qProof.
(5)Qv−1MBαNTlqu,v=q2u3v2+qu3v4+qu3v5+3q2u4v3+2qu4v4+qu4v5+q23l−8u5v4+q2l−8u5v5,Qu−1Qv−1MBαNTlqu,v=q2u2v2+qu2v4+qu2v5+3q2u3v3+2qu3v4+qu3v5+q23l−8u4v4+q2l−8u4v5,Dv1/2Qu−1Qv−1MBαNTlqu,v=22qu2v2+2qu2v4+5qu2v5+332qu3v3+4qu3v4+5qu3v5+q3l−8u4v4+5q2l−8u4v5,(6)Du1/2Dv1/2Qu−1Qv−1MBαNTlqu,v=qu2v2+22qu2v4+10qu2v5+92qu3v3+43qu3v4+15qu3v5+2q3l−8u4v4+25q2l−8u4v5,Sv1/2MBαNTlqu,v=36qu3v3+55qu3v5+66qu3v6+34qu4v4+255qu4v5+66qu4v6+510q3l−8u5v5+66q2l−8u5v6,Su1/2Sv1/2MBαNTlqu,v=16qu3v3+1515qu3v5+26qu3v6+38qu4v4+55qu4v5+612qu4v6+110q3l−8u5v5+3030q2l−8u5v6,(7)JSu1/2Sv1/2MBαNTlqu,v=16qu6+112045+815qu8+13052+65qu9+612qu10+110q3l−8u10+3030q2l−8u11,DuJSu1/2Sv1/2MBαNTlqu,v=qu6+11545+815qu8+31052+65qu9+566qu10+q3l−8u10+113030q2l−8u11,12DuJSu1/2Sv1/2MBαNTlqu,v=12qu6+13045+815qu8+32052+65qu9+5612qu10+12q3l−8u10+113060q2l−8u11,(8)DuMBαNTlqu,v=32qu3v3+3qu3v5+3qu3v6+6qu4v4+8qu4v5+4qu4v6+52q3l−8u5v5+5q2l−8u5v6,DvMBαNTlqu,v=32qu3v3+5qu3v5+6qu3v6+6qu4v4+10qu4v5+6qu4v6+52q3l−8u5v5+6q2l−8u5v6,Du+DvMBαNTlqu,v=3qu3v3+8qu3v5+9qu3v6+12qu4v4+18qu4v5+10qu4v6+5q3l−8u5v5+11q2l−8u5v6,(9)12Du+DvMBαNTlqu,v=32qu3v3+4qu3v5+92qu3v6+6qu4v4+9qu4v5+5qu4v6+52q3l−8u5v5+112q2l−8u5v6,DuDvMBαNTlqu,v=92qu3v3+15qu3v5+18qu3v6+24qu4v4+40qu4v5+24qu4v6+252q3l−8u5v5+30q2l−8u5v6,12DuDvMBαNTlqu,v=94qu3v3+152qu3v5+9qu3v6+12qu4v4+20qu4v5+12qu4v6+254q3l−8u5v5+15q2l−8u5v6,(10)JMBαNTlqu,v=q2u6+52qu8+3qu9+q23l−6u10+q2l−8u11,Du2JMBαNTlqu,v=18qu6+160qu8+243qu9+50q3l−6u10+121q2l−8u11,14Di2JMBαNTlqu,v=92qu6+40qu8+2434qu9+252q3l−6u10+1214q2l−8u11,Qu−2JMBαNTlqu,v=q2u4+52qu6+3qu7+q23l−6u8+q2l−8u9,(11)Du2Qu−2JMBαNTlqu,v=8qu4+90qu6+147qu7+32q3l−6u8+81q2l−8u9,Su1/2JMBαNTlqu,v=612qu6+528qu8+qu9+31010q3l−6u10+1111q2l−8u11,DuλJMBαNTlqu,v=6λ2qu6+5⋅8λ2qu8+3⋅9λqu9+10λ2q3l−6u10+11λq2l−8u11,DvDu+DvMBαNTlqu,v=9qu3v3+40qu3v5+54qu3v6+48qu4v4+90qu4v5+60qu4v6+253l−8u5v5+66q2l−8u5v6,DuDvDu+DvMBαNTlqu,v=27qu3v3+120qu3v5+162qu3v6+192qu4v4+360qu4v5+240qu4v6+1253l−8u5v5+330q2l−8u5v6.(1)
RRRBαNTlq =Du1/2Dv1/2Qu−1Qv−1MBαNTlqu,v|u=v=1,RRRBαNTlq=6+45ql−1/221−42−83+205−210−215q.(2)
AG1BαNTlq=1/2DuJSu1/2Sv1/2MBαNTlqu,v|u=1,AG1BαNTlq=1/3045+1130ql+1/60452+545+256+1612−5530−120q.(3)
SKBαNTlq=1/2Du+DvMBαNTlqu,v|u=v=1=37/2ql−35/2q.(4)
SK1BαNTlq =1/2DuDvMBαNTlqu,v|u=v=1=195/4ql−249/4q.(5)
Sk2BαNTlq=1/4Du2JMBαNTlqu,v|u=1=98ql−121q.(6)
EM1BαNTlq =Du2Qu−2JMBαNTlqu,v|u=1=258ql−352q.(7)
SCIBαNTlq =Su1/2JMBαNTlqu,v|u=1,SCIBαNTlq=1/2203310+4011ql+1/13201320+8252+1106−39610−60011q.(8)
SCIλBαNTlq =DuλJMBαNTlqu,v|u=1,SCIλBαNTlq=1/23⋅10λ+4⋅11λql+1/26λ+5⋅8λ+6⋅9λ−6⋅10λ−10⋅11λq.(9)
ReZG3BαNTlq =DuDvDu+DvMBαNTlqu,v|u=v=1=1035ql−1549q.


## 4. Conclusion

We studied the important nanotube known as boronα-nanotube by computing their M-polynomial and then recovered some important degree-based TIs. The graphical representations of the results are presented in Figure 3. The results obtained will be useful in helping to solve many problems in the field of chemical analysis.Figure 3
Plot of topological indices of boronα-nanotube: (a) reduced reciprocal Randić index; (b) arithmetic-geometric index; (c) SK index; (d) SK1 index; (e) SK2 index; (f) edge version of first Zagreb index; (g) sum connectivity index; (h) general sum connectivity index; (i) redefined third Zagreb index; (j) TI of BαNTlq|q=4; (k) TI of BαNTlq|l=4; (l) colour of TI.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)



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*Source: 6136168-2022-01-28.xml*