# On Computation of Recently Defined Degree-Based Topological Indices of Some Families of Convex Polytopes viaM-Polynomial

Authors: Deeba Afzal; Farkhanda Afzal; Mohammad Reza Farahani; Samia Ali
Journal: Complexity
 (2021)

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## Abstract

Topological indices are of incredible significance in the field of graph theory. Convex polytopes play a significant role both in various branches of mathematics and also in applied areas, most notably in linear programming. We have calculated some topological indices such as atom-bond connectivity index, geometric arithmetic index,K-Banhatti indices, and K-hyper-Banhatti indices and modified K-Banhatti indices from some families of convex polytopes through M-polynomials. The M-polynomials of the graphs provide us with a great help to calculate the topological indices of different structures.


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## Body

## 1. Introduction

Graph theory is a powerful and definable field of mathematics that in every field of science has countless adaptations. In graph theory, different characteristics of graphs are defined [1]. The topological indices display the graphical structure and many other characteristics in graphs. They are typically based on the distances between the vertices, on vertex degrees, or on the graph depicted by the matrix.Using the general polynomial is the general method by which we can generate the unique form of topological indices. Hosoya polynomial is an example of this form [2]. Similarly M-polynomial [3] that is an algebraic polynomial can also explain the behavior of the molecular structure. It is also graph representative mathematical object which is the most general polynomial developed till now and gives us the formulas that are very close to the degree-based topological indices.TheM-polynomial introduced in 2015 for a graph G by Emeric Deutsch and Sandi Klavžar [4] is defined as(1)MG,x,y=∑δ≤i≤j≤ΔmijGxiyj.Here,mijG is the number of edges uv∈EG, such that dv,du=i,j, δ=mindv|v∈VG, and Δ=maxdv|v∈VG.We generally find topological indices using the definitions, but we can also find them using the chosen graph’s derivatives and integrals onM-polynomials. So if we have an M-polynomial of the graph, we can measure its various topological indices. Several M-polynomials of several graphs have been added over the last few years [3, 5, 6].In 1947, first index, the wiener index suggested by Wiener. A well-known degree-based topological invariant known as atom-bond connectivity index [7] was purposed in 1998, by Estrada et al. This index is denoted by ABC. Damir Vukičević and Boris Furtula suggested a new index in 2009, known as geometric arithmetic index [8] which is denoted by GA. K-Banhatti and K-hyper-Banhatti indices were introduced by Kulli in 2016 [9, 10]. Similarly, in 2017, Kulli introduced some more important topological indices [11] named as modified K-Banhatti indices. Here we will use the operators which are defined as(2)Dxgx,y=x∂gx,y∂x,Dygx,y=y∂gx,y∂y,Lxgx,y=gx2,y,Lygx,y=gx,y2,Sxgx,y=∫0xgt,ytdt,Sygx,y=∫0ygx,ttdt,Jgx,y=gx,x,Qαgx,y=xαgx,y,Dx1/2gx,y=x∂gx,y∂x⋅gx,y,Dy1/2gx,y=y∂gx,y∂y⋅gx,y,Sx1/2gx,y=∫0xgt,ytdt⋅gx,y,Sy1/2gx,y=∫0ygx,ttdt⋅gx,y.Now we will calculate closed forms of many degree-based topological indices of some families of convex polytopes by usingM-polynomials.


## 2. Some Families of Convex Polytopes and Their Computational Aspects

If we add new edges in the graph of convex polytopesQn, then we get a graph named as convex polytopes Tn. It will consist of three-sided face, five-sided face, and n-sided faces. ai+1bi, i.e., VTn=VQn and VTn and VQn∪ai+1bi|l≤i≤n (for example, see Figure 1).Figure 1
Graph of convex polytopeT6.Similarly, the graph of convex polytopeAn, i.e., double antiprism, can be acquired from the graph of convex polytopes Rn by adding new edges bi+1ci, i.e., VAn=VRn and VAn=Rn∪bi=lci|l≤i≤n (for example, see Figure 2).Figure 2
Graph of double antiprismA6.Convex polytopes (double antiprism)Sn graph can be obtained from the graph of convex poly tope Qn by adding new edges cici+1, i.e., VSn=VQn and VSn=VQn∪cici+1|l≤i≤n (for example, see Figure 3).Figure 3
Graph of double antiprismS6.


## 3.M-Polynomial and Topological Indices of Convex Polytopes Tn

Now we will find the topological indices ofTn using its M-polynomial.Theorem 1.
TheM-polynomial of Tn is MTn;x;y=2nx3y3+2nx3y6+nx4y4+2nx4y6+nx6y6. Then(1)
ABCTn=n/1216+414+36+48+10(2)
GATn=n/1530+102+66(3)
B1Tn=178n(4)
B2Tn=502n(5)
HB1Tn=2126n(6)
HB2Tn=2518n(7)
B1mTn=17033/10920n(8)
B2mTn=1171/1680n(9)
HbTn=17033/5460n.Proof.
(3)Dxgx;y=6nx3y3+6nx3y6+4nx4y4+8nx4y6+6nx6y6;Dygx;y=6nx3y3+12nx3y6+4nx4y4+12nx4y6+6nx6y6;Dx+Dygx;y=12nx3y3+18nx3y6+8nx4y4+20nx4y6+12nx6y6;Sy1/2gx,y=23nx3y3+26nx3y6+n2x4y4+26nx4y6+n2x6y6,Sx1/2Sy1/2gx,y=23nx3y3+232nx3y6+n4x4y4+16nx4y6+n6x6y6,JSx1/2Sy1/2gx,y=23nx6+232nx9+n4x8+16nx10+n6x12,Q−2JSx1/2Sy1/2gx,y=23nx4+232nx7+n4x6+16nx8+n6x10,Dx1/2Q−2JSx1/2Sy1/2gx,y=43nx4+143nx7+64nx6+86nx8+106nx10,Dy1/2gx,y=23nx3y3+26nx3y6+2nx4y4+26nx4y6+n6x6y6,Dx1/2Dy1/2gx,y=6nx3y3+263nx3y6+4nx4y4+246nx4y6+6nx6y6,JDx1/2Dy1/2gx,y=6nx6+263nx9+4nx8+246nx10+6nx12,SxJDx1/2Dy1/2gx,y=nx6+232nx9+12nx8+256nx10+12nx12,2SxJDx1/2Dy1/2gx,y=2nx6+432nx9+nx8+456nx10+nx12,Jgx;y=2nx6+nx8+2nx9+2nx10+nx12;Q−2Jgx;y=2nx4+nx6+2nx7+2nx8+nx10;DxQ−2Jgx;y=8nx4+6nx6+14nx7+16nx8+10nx10;2DxQ−2Jgx;y=16nx4+12nx6+28nx7+32nx8+20nx10;JDx+Dygx,y=12nx6+18nx9+8nx8+20nx10+12nx12;Q−2JDx+Dygx;y=12nx4+18nx7+8nx6+20nx8+12nx10;DxQ−2JDx+Dygx;y=48nx4+126nx7+48nx6+160nx8+120nx10;Dx2gx;y=18nx3y3+18nx3y6+16nx4y4+32nx4y6+36nx6y6;Dy2gx;y=18nx3y3+72nx3y6+16nx4y4+72nx4y6+36nx6y6;Dx2Q−2Jgx;y=32nx4+98nx7+36nx6+128nx8+100nx10;2D2Q−2Jgx;y=64nx4+196nx7+72nx6+256nx8+200nx10;2DxQ−2JDx+Dygx;y=96nx4+252nx7+96nx6+320nx8+240nx10;Dx2+Dy2gx;y=36nx3y3+90nx3y6+32nx4y4+104nx4y6+72nx6y6;JDx2+Dy2gx;y=36nx6+90nx9+32nx8+104nx10+72nx12;Q−2JDx2+Dy2gx;y=36nx4+90nx7+32nx6+104nx8+72nx10;Dx2Q−2JDx2+Dy2gx;y=576nx4+4410nx7+1152nx6+6656nx8+7200nx10;Lxgx;y=2nx6y3+2nx6y6+nx8y4+2nx8y6+nx12y6;Lygx;y=2nx3y6+2nx3y12+nx4y8+2nx4y12+nx6y12;JLx+Lygx;y=4nx9+4nx12+2nx14+2nx15+2nx16+2nx18;Q−2JLx+Lygx;y=4nx7+4nx10+2nx12+2nx13+2nx14+2nx16;SxQ−2JLx+Lygx,y=47nx7+410nx10+16nx12+213nx13+17nx14+18nx16,Sygx,y=23nx3y3+13nx3y6+14nx4y4+13nx4y6+16nx6y6,Sxgx,y=23nx3y3+23nx3y6+14nx4y4+12nx4y6+16nx6y6,Sx+Sygx,y=43nx3y3+nx3y6+12nx4y4+56nx4y6+13nx6y6,JSx+Sygx,y=43nx6+nx9+12nx8+56nx10+13nx12,Q−2JSx+Sygx,y=43nx4+nx7+12nx6+56nx8+13nx10,SxQ−2JSx+Sygx,y=13nx4+n7x7+112nx6+548nx8+130nx10,2SxQ−2JLx+Lygx,y=87nx7+45nx10+13nx12+413nx13+27nx14+14nx16.
Hence, we conclude the following results:(1)
ABCTn=Dx1/2Q−2JSx1/2Sy1/2gx,yx=1=n/1216+414+36+48+10(2)
GATn=2SxJDx1/2Dy1/2gx,yx=1=n/1530+102+66(3)
B1Tn=Dx+Dy+2DxQ−2Jgx,yx=y=1=178n(4)
B2Tn=DxQ−2JDx+Dygx,yx=1=502n(5)
HB1Tn=Dx2+Dy2+2Dx2Q−2J+2DxQ−2JDx+Dygx,yx=y=1=2126n(6)
HB2Tn=Dx2Q−2JDx2+Dy2gx,yx=1=19994n(7)
B1mTn=SxQ−2JLx+Lygx,yx=1=17033/10920n(8)
B2mTn=SxQ−2JSx+Sygx,yx=1=1171/1680n(9)
HbTn=2SxQ−2JLx+Lygx,yx=1=17033/5460n.


## 4.M-Polynomial and Topological Indices of Convex Polytopes An

Here, we will find the topological indices ofAn using its M-polynomial.Theorem 2.
TheM-polynomial of An is(4)MAn;x;y=2nx4y4+4nx4y6+nx6y6.Then,(1)
ABCAn=n/36186+483+610(2)
GAAn=n/515+86(3)
B1An=176n(4)
B2An=536n(5)
HB1An=2272n(6)
HB2An=2768n(7)
B1mAn=961/840n(8)
B2mAn=49/120n(9)
HbAn=961/420n.Proof.
(5)Dxgx;y=8nx4y4+16nx4y6+6nx6y6;Dygx;y=8nx4y4+24nx4y6+6nx6y6;Dx+Dygx;y=16nx4y4+40nx4y6+12nx6y6;Sy1/2gx,y=nx4y4+46nx4y6+n6x6y6,Sx1/2Sy1/2gx,y=n2x4y4+46nx4y6+n6x6y6,JSx1/2Sy1/2gx,y=n2x8+46nx10+n6x12,Q−2JSx1/2Sy1/2gx,y=n2x6+46nx8+n6x10,Dx1/2Q−2JSx1/2Sy1/2gx,y=32nx6+413nx8+106nx10,Dy1/2gx,y=4nx4y4+46nx4y6+6nx6y6,Dx1/2Dy1/2gx,y=8nx4y4+86nx4y6+6nx6y6,JDx1/2Dy1/2gx,y=8nx8+86nx10+6nx12,SxJDx1/2Dy1/2gx,y=nx8+456nx10+12nx12,2SxJDx1/2Dy1/2gx,y=2nx8+856nx10+nx12,Jgx;y=2nx8+4nx10+nx12;Q−2Jgx;y=2nx6+4nx8+nx10;DxQ−2Jgx;y=12nx6+32nx8+10nx10;2DxQ−2Jgx;y=24nx6+64nx8+20nx10;JDx+Dygx;y=16nx8+40nx10+12nx12;Q−2JDx+Dygx;y=16nx6+40nx8+12nx10;DxQ−2JDx+Dygx;y=96nx6+320nx8+120nx10;Dx2gx;y=32nx4y4+64nx4y6+36nx6y6;Dy2gx;y=32nx4y4+144nx4y6+36nx6y6;Dx2+Dy2gx;y=64nx4y4+208nx4y6+72nx6y6;Dx2Q−2Jgx;y=72nx6+256nx8+100nx10;2Dx2Q−2Jgx;y=144nx6+512nx8+200nx10;2DxQ−2JDx+Dygx;y=192nx6+640nx8+240nx10;JDx2+Dy2gx;y=64nx8+208nx10+72nx12;Q−2JDx2+Dy2gx;y=64nx6+208nx8+72nx10;Dx2Q−2JDx2+Dy2gx;y=384nx6+1664nx8+720nx10;Lxgx;y=2nx8y4+4nx8y6+nx12y6;Lygx;y=2nx4y8+4nx4y12+nx6y12;JLx+Lygx;y=4nx12+4nx16+4nx14+2nx18;Q−2JLx+Lygx;y=4nx10+4nx14+4nx12+2nx16;SxQ−2JLx+Lygx,y=25nx10+27nx14+13nx12+18nx16,Sygx,y=12nx4y4+23nx4y6+n6nx6y6,Sxgx,y=12nx4y4+nx4y6+n6nx6y6,Sx+Sygx,y=nx4y4+53nx4y6+n3nx6y6,JSx+Sygx,y=nx8+53nx10+n3nx12,Q−2JSx+Sygx,y=nx6+53nx8+n3nx10,SxQ−2JSx+Sygx,y=n6x6+524nx8+n30nx10,2SxQ−2JLx+Lygx,y=45nx10+47nx14+23nx12+14nx16.
Hence, we conclude the following results:(1)
ABCAn=Dx1/2Q−2JSx1/2Sy1/2gx,yx=1=n/36186+483+610(2)
GAAn=2SxJDx1/2Dy1/2gx,yx=1=n/515+86(3)
B1An=Dx+Dy+2DxQ−2Jgx,yx=y=1=176n(4)
B2An=DxQ−2JDx+Dygx,yx=1=536n(5)
HB1An=Dx2+Dy2+2Dx2Q−2J+2DxQ−2JDx+Dygx,yx=y=1=2272n(6)
HB2An=Dx2Q−2JDx2+Dy2gx,yx=1=2768n(7)
B1mAn=SxQ−2JLx+Lygx,yx=1=961/840n(8)
B2mAn=SxQ−2JSx+Sygx,yx=1=49/120n(9)
HbAn=2SxQ−2JLx+Lygx,yx=1=961/420n.


## 5.M-Polynomial and Topological Indices of Convex Polytopes Sn

Now we will find the topological indices ofSn using its M-polynomial.Theorem 3.
TheM-polynomial of Sn is(6)MSn;x;y=2nx3y3+2nx3y5+4nx5y5.Then,(1)
ABCSn=n/1520+610+242(2)
GASn=n/212+15(3)
B1Sn=172n(4)
B2Sn=464n(5)
HB1Sn=15824n(6)
HB2Sn=1952(7)
B1mSn=14332/9009(8)
B2mSn=32/45n(9)
HbSn=28664/9009n.Proof.
(7)Dxgx,y=6nx3y3+6nx3y5+20nx5y5,Dygx,y=6nx3y3+10nx3y5+20nx5y5,Dx+Dygx,y=12nx3y3+16nx3y5+40nx5y5,Sy1/2gx,y=23nx3y3+25nx3y5+45nx5y5,Sx1/2Sy1/2gx,y=23nx3y3+215nx3y5+45nx5y5,JSx1/2Sy1/2gx,y=23nx6+215nx8+45nx10,Q−2JSx1/2Sy1/2gx,y=23nx4+215nx6+45nx8,Dx1/2Q−2JSx1/2Sy1/2gx,y=43nx4+2615nx6+825nx8,Dy1/2gx,y=23nx3y3+25nx3y5+45nx5y5,Dx1/2Dy1/2gx,y=6nx3y3+215nx3y5+20nx5y5,JDx1/2Dy1/2gx,y=6nx6+215nx8+20nx10,SxJDx1/2Dy1/2gx,y=nx6+154nx8+2nx10,2SxJDx1/2Dy1/2gx,y=2nx6+152nx8+4nx10,Jgx,y=2nx6+2nx8+4nx10,Q−2Jgx,y=2nx4+2nx6+4nx8,DxQ−2Jgx,y=8nx4+12nx6+32nx8,2DxQ−2Jgx,y=16nx4+24nx6+64nx8,JDx+Dygx,y=12nx6+16nx8+40nx10,Q−2JDx+Dygx,y=12nx4+16nx6+40nx8,DxQ−2JDx+Dygx,y=48nx4+96nx6+320nx8,Dx2gx,y=18nx3y3+18nx3y5+100nx5y5,Dy2gx,y=18nx3y3+50nx3y5+100nx5y5,Dx2+Dy2gx,y=36nx3y3+68nx3y5+200nx5y5,Dx2Q−2Jgx,y=32nx4+72nx6+256nx8,2Dx2Q−2Jgx,y=64nx4+144nx6+512nx8,2DxQ−2JDx+Dygx,y=96nx4+192nx6+640nx8,JDx2+Dy2gx,y=36nx6+68nx8+200nx10,Q−2JDx2+Dy2gx,y=36nx4+68nx6+200nx8,Dx2Q−2JDx2+Dy2gx,y=576nx4+2448nx6+12800nx8,Lxgx,y=2nx6y3+2nx6y5+4nx10y5,Lygx,y=2nx3y6+2nx3y10+4nx5y10,JLx+Lygx,y=4nx9+8nx15+2nx13+2nx11,Q−2JLx+Lygx,y=4nx7+8nx13+2nx11+2nx9,SxQ−2JLx+Lygx,y=47nx7+813nx13+211nx11+29nx9,Sygx,y=12nx4y4+23nx4y6+n6x6y6,Sxgx,y=12nx4y4+nx4y6+n6x6y6,Sx+Sygx,y=nx4y4+53nx4y6+n3x6y6,JSx+Sygx,y=nx8+53nx10+n3x12,Q−2JSx+Sygx,y=nx6+53nx8+n3x10,SxQ−2JSx+Sygx,y=n6x6+524nx8+n30x10,2SxQ−2JLx+Lygx,y=87nx7+1613nx13+411nx11+49nx9.
Hence, we conclude the following results:(1)
ABCSn=Dx1/2Q−2JSx1/2Sy1/2gx,yx=1=n/1520+610+242(2)
GASn=2SxJDx1/2Dy1/2gx,yx=1=n/212+15(3)
B1Sn=Dx+Dy+2DxQ−2Jgx,yx=y=1=172n(4)
B2Sn=DxQ−2JDx+Dygx,yx=1=464n(5)
HB1Sn=Dx2+Dy2+2Dx2Q−2J+2DxQ−2JDx+Dygx,yx=y=1=15824n(6)
HB2Sn=Dx2Q−2JDx2+Dy2gx,yx=1=1952n(7)
B1mSn=SxQ−2JLx+Lygx,yx=1=14332/9009n(8)
B1mSn=SxQ−2JSx+Sygx,yx=1=32/45n(9)
HbSn=2SxQ−2JLx+Lygx,yx=1=28664/9009n.


## 6. Graphical Analysis

Graphs are helpful as they elucidate the facts in visual form. Moreover, we demonstrate the behavior ofM-polynomial and topological indices by drawing graphics. A graph stores information of the chemical structures that is equal to several words. The information that could not be represented in terms of words can be clarified in a better way by means of a graph. This graphical description provides us a great way to understand results against parameters (see Figures 4–6 ).Figure 4
The plot of topological indices of convex polytopeTn. (a) Atom-bond connectivity index, (b) geometric arithmetic index, (c) first K-Banhatti index, (d) second K-Banhatti index, (e) first K-hyper-Banhatti index, (f) second K-hyper-Banhatti index, (g) modified first K-Banhatti index, (h) modified second K-Banhatti index, and (i) harmonic K-Banhatti index.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Figure 5
The plot of topological indices of convex polytopeAn. (a) Atom-bond connectivity index, (b) geometric arithmetic index, (c) first K-Banhatti index, (d) second K-Banhatti index, (e) first K-hyper-Banhatti index, (f) second K-hyper-Banhatti index, (g) modified first K-Banhatti index, (h) modified second K-Banhatti index, and (i) harmonic K-Banhatti index.
(a)(b)(c)(d)(e)(f)(g)(h)(i)Figure 6
The plot of topological indices of convex polytopeSn. (a) Atom-bond connectivity index, (b) geometric arithmetic index, (c) first K-Banhatti index, (d) second K-Banhatti index, (e) first K-hyper-Banhatti index, (f) second K-hyper-Banhatti index, (g) modified first K-Banhatti index, (h) modified second K-Banhatti index, and (i) harmonic K-Banhatti index.
(a)(b)(c)(d)(e)(f)(g)(h)(i)


## 7. Conclusion

Topological indices are a numerical quantity computed from the graph of any compound and typically remain the same under graph isomorphism. These are used to predict and model certain properties of different compounds. We can the easily understand the properties, for example, the chemical reactivity, biological activities, and physical features of associated molecules. In this work, we have calculated very important degree-based topological indices such as atom-bond connectivity index, geometric arithmetic index,K-Banhatti indices, K-hyper-Banhatti indices and modified K-Banhatti indices of some families of cycle graphs which are given in Table 1 with the help of their M-polynomials. These findings can play a huge role in industry and pharmacy because these indices are absolutely functions of chemical graphs and explain many chemical properties as viscosity, strain energy, and heat of formation etc.Table 1
Topological indices derived fromMG;x;y=gx;y[12].
ABCG=Dx1/2Q−2JSx1/2Sy1/2gx,yx=1GAG=2SxJDx1/2Dy1/2gx,yx=1B1G=Dx+Dy+2DxQ−2Jgx,yx=y=1B2G=DxQ−2JDx+Dygx,yx=1HB1G=Dx2+Dy2+2Dx2Q−2J+2DxQ−2JDx+Dygx,yx=y=1HB2G=Dx2Q−2JDx2+Dy2gx,yx=1B1mG=SxQ−2JLx+Lygx,yx=1B2mG=SxQ−2JSx+Sygx,yx=1HbG=2SxQ−2JLx+Lygx,yx=1



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*Source: 5881476-2021-07-08.xml*