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# Open Chemistry

### formerly Central European Journal of Chemistry

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Volume 16, Issue 1

# M-Polynomials and Topological Indices of Dominating David Derived Networks

Shin Min Kang
• Corresponding author
• Department of Mathematics and RINS, Gyeongsang National University, Jinju, 52828, Korea
• Center for General Education, China Medical University, Taichung, 40402, Taiwan
• Email
• Other articles by this author:
/ Waqas Nazeer
/ Wei Gao
• School of Information Science and Technology, Yunnan Normal University, Kunming, 650500, China
• Other articles by this author:
/ Deeba Afzal
/ Syeda Nausheen Gillani
Published Online: 2018-03-20 | DOI: https://doi.org/10.1515/chem-2018-0023

## Abstract

There is a strong relationship between the chemical characteristics of chemical compounds and their molecular structures. Topological indices are numerical values associated with the chemical molecular graphs that help to understand the physical features, chemical reactivity, and biological activity of chemical compound. Thus, the study of the topological indices is important. M-polynomial helps to recover many degree-based topological indices for example Zagreb indices, Randic index, symmetric division idex, inverse sum index etc. In this article we compute M-polynomials of dominating David derived networks of the first type, second type and third type of dimension n and find some topological properties by using these M-polynomials. The results are plotted using Maple to see the dependence of topological indices on the involved parameters.

## 1 Introduction

The David derived and dominating David derived network of dimension n can be constructed as follows [1]: consider a n dimensional star of David network SD(n) [4]. Insert a new vertex on each edge and split it into two parts, this gives the David derived network DD(n of dimension n. The dominating David derived network of the first type of dimension n which can be obtained by connecting vertices of degree 2 of DDD(n) by an edge that are not in the boundary and is denoted by D1 (n) [1].

The dominating David derived network of the second type of dimension n can be obtained by subdividing the new edges in D1 (n) [1] and is denoted by D1 (2).

where $Dx=x∂(f(x,y)∂x,Dy=y∂(f(x,y)∂y,Sx=∫0χf(t,y)tdt,Sy==∫0yf(x,t)tdt,J(f(x,y))=f(x,x),Qα(f(x,y))=xaf(x,y).$

The dominating David derived network of the second type of dimension n denoted by D3 (n) can be obtained from D1 (n) by introducing a parallel path of length 2 between the vertices of degree two that are not in the boundary [1, 5].

In this report, M-polynomials of dominating David derived networks of the first type, second type and third type of dimension n, are computed. From these M-polynomials many degree-based topological indices are recovered. For example: first Zagreb index, second Zagreb index, modified second Zagreb index, Symmetric division index, generalized Randić index, generalized inverse Randić index, Augmented Zagreb index, etc for underlined networks. The results are plotted using maple 2015 software.

For basic definitions see [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

The following table 1 relates some well-known degreebased topological indices to M-polynomial [2].

Table 1

Derivation of some degree-based topological indices from M-polynomial.

More results on the computation of these indices can refer to [20, 21, 22, 23, 24, 25, 26, 27].

Ethical approval: The conducted research is not related to either human or animals use.

## 2.1 M-Polynomial and topological indices of dominating David derived network of first type

Let G = D1 (n) be the dominating David derived network of first type. From Figure 1, this gives vertex and edge partitions (see Table 2,3)

Figure 1

Dominating David derived network of the first type D1 (2).

Table 2

The vertex partition of set of D1 (n).

Table 3

The Edge partition of.

#### Theorem 1

Let D1(n) be the dominating David derived network of first type. Then the M-Polynomial of D1(n) is M (D1 (n);x,y) = 4nx2y2+ (4n − 4)x2y3 +(28n − 16)x2y4 + + (9n2 − 3n + 5)x3y3 + (36n2 − 56n + 24)x3x4 + (36n2 − 52n + 20)x4 y4

#### Proof

Let D1(n) is the dominating David derived network of first type. It is easy to see form Figure 1 that

From Table 2, the vertex set of D1(n) have three partitions: $V1(D1(n))={u∈V(D1(n)):du=2},V2(D1(n))={u∈V(D1(n)):du=3},V3(D1(n))={u∈V(D1(n)):du=4},$

such that |V1(D1(n))| = 20n − 10, |V2(D1(n))| = 18n2 − 26n +10, and |V3(D1(n))| = 27n2 − 33n +12 .

From Table 3, the edge set of D1(n) have six partitions: $E1(D1(n))={e=uv∈E(D1(n)):du=2,dv=2},E2(D1(n))={e=uv∈E(D1(n)):du=2,dv=3},E3(D1(n))={e=uv∈E(D1(n)):du=2,dv=4},E4(D1(n))={e=uv∈E(D1(n)):du=3,dv=3},E5(D1(n))={e=uv∈E(D1(n)):du=3,dv=4},E6(D1(n))={e=uv∈E(D1(n)):du=4,dv=4}.$

By means of Figure 1, this gives $|E1(D1(n))|=4n,|E2(D1(n))|=4n−4,|E3(D1(n))|=28n−16,|E4(D1(n))|=9n2−13n+5,|E5(D1(n))|=36n2−56n+24,|E6(D1(n))|=36n2−52n+20.$

Now according to the definition of the M-polynomial, this gives $M(D1(n);x,y)=∑mijxjyj=∑uv∈E1(D1(n))m22x2y2i≤j+∑uv∈E2(D1(n))m23x2y3+∑uv∈E3(D1(n))m24x2y4+∑uv∈E4(D1(n))m33x3y3+∑uv∈E5(D1(n))m34x3y4+∑uv∈E6(D1(n))m44x4y4=|E1(D1(n))|x2y2+|E2(D1(n))|x2y3+|E3(D1(n))|x2y4+|E4(D1(n))|x3y3+|E5(D1(n))|x3y4+|E6(D1(n))|x4y4=4nx2y3+(4n−4)x2y3+(28n−16)x2y4+(9n2−13n+5)x3y3+(36n2−56n+24)x3y4+(36n2−52n+20)x4y4.$

Figure 4 (shown above) is plotted by using Maple 15. This suggests that values obtained by M-polynomial show different behaviors corresponding to different parameters x and y. The values of M-polynomial can be controlled through these parameters. Clearly Figure 4 shows that along one side intercept is an upward opening parabola.

Figure 2

Dominating David derived network of the second type D2 (4).

Figure 3

Dominating David derived network of the third type D3 (2).

Figure 4

Plot of M-Polynomial of Dominating David Derived Network of first Type.

Next some degree-based topologcal indices of dominating David derived network of first type are computed from this M-polynomial.

#### Proposition 2

Let D1(n) be the dominating David derived network of first type

1. M1(D1(n)) = 594n2 − 682n + 242

2. M2(D1(n)) = 1089n2 − 1357n + 501.

3. $\begin{array}{}{}^{m}{M}_{2}\left({D}_{1}\left(n\right)\right)=\frac{25}{4}{n}^{2}-\frac{151}{36}n+\frac{41}{36}.\end{array}$

4. Rα (D1(n)) = 22 α 4n + 6α (4n − 4) + 8α (28n − 16) + 32α (9n2 − 13n + 5) + + 12α (36n2 − 56n + 24) +42α (36n2 − 52n + 20).

5. $\begin{array}{}R{R}_{\alpha }\left({D}_{1}\left(n\right)\right)=\frac{1}{{2}^{2\alpha -2}}n+\frac{1}{{6}^{\alpha }}\left(4n-4\right)+\frac{1}{{2}^{3\alpha }}\left(28n-16\right)+\frac{1}{{3}^{2\alpha }}\left(9{n}^{2}-13n+5\\ \frac{1}{{12}^{\alpha }}\left(36{n}^{2}-56n+20\right)+\frac{1}{{2}^{4\alpha }}\left(36{n}^{2}-52n+20\right).\end{array}$

6. SSD(D1(n)) = 165n2 − 160n + $\begin{array}{}\frac{154}{3}.\end{array}$

7. $\begin{array}{}H\left({D}_{1}\left(n\right)\right)=\frac{156}{7}{n}^{2}-\frac{102}{5}n+\frac{692}{105}.\end{array}$

8. $\begin{array}{}I\left({D}_{1}\left(n\right)\right)=\frac{2061}{14}{n}^{2}+\frac{5201}{30}n+\frac{13127}{210}.\end{array}$

9. $\begin{array}{}A\left({D}_{1}\left(n\right)\right)=\frac{30788311}{24000}{n}^{2}-\frac{1858527979}{216000}n+\frac{130461491}{216000}.\end{array}$

#### Proof

Let $f(x,y)=4nx2y2+(4n−4)x2y3+(28n−16)x2y4+(9n2−3n+5)x3y3+(36n2−56n+24)x3y4+(36n2−52n+20)x4y4.$

Then $Dx(f(x,y))=8nx2y2+2(4n−4)x2y3+2(28n−16)x2y4+3(9n2−13n+5)x3y3+3(36n2−52n+24)x3y4+4(36n2−52n+20)x4y4.Dy(f(x,y))=8nx2y2+3(4n−4)x2y3+4(28n−16)x2y4+3(9n2−13n+5)x3y3+4(36n2−52n+24)x3y4+4(36n2−52n+20)x4y4.(DxDy)(f(x,y))=16nx2y2+6(4n−4)x2y3+8(28n−16)x2y4+9(9n2−13n+5)x3y3+12(36n2−52n+24)x3y4+16(36n2−52n+20)x4y4,SxSy(f(x,y))=nx2y2+16(4n−4)x2y3+18(28n−16)x2y4+19(9n2−13n+5)x3y3+112(36n2−52n+24)x3y4+116(36n2−52n+20)x4y4DxaDya(f(x,y))=22a4nx2y2+6a(4n−4)x2y3+8a(28n−16)x2y4+32a(9n2−13n+5)x3y3+12a(36n2−52n+24)x3y4+42a(36n2−52n+20)x4y4,SxaSya(f(x,y))=122a−2nx2y2+16a(4n−4)x2y3+18a(28n−16)x2y4+132a(9n2−13n+5)x3y3+112a(36n2−52n+24)x3y4+142a(36n2−52n+20)x4y4,DxSy(f(x,y))=4nx2y2+23(4n−4)x2y3+12(28n−16)x2y4+(9n2−3n+5)x3y3+34(36n2−56n+24)x3y4+(36n2−52n+20)x4y4,SxDy(f(x,y))=4nx2y2+32(4n−4)x2y3+2(28n−16)x2y4+(9n2−3n+5)x3y3+43(36n2−56n+24)x3y4+(36n2−52n+20)x4y4,SxJf(x,y)=nx4+15(4n−4)x5+16(28n−16)x6+16(9n2−3n+5)x6+17(36n2−56n+24)x7+18(36n2−52n+20)x8,SxJDxDy(f(x,y))=4nx4+65(4n−4)x5+43(28n−16)x6+32(9n2−3n+5)x6+127(36n2−56n+24)x7+2(36n2−52n+20)x8,Sx3Q−2JDx3Dy3f(x,y)=32nx2+8(4n−4)x3+8(28n−16)x4+72964(9n2−3n+5)x4+1728125(36n2−56n+24)x5+51227(36n2−52n+20)x6.$

Now in view of Table 1, this gives:

1. M1(D1(n)) = (Dx + Dy)(f (x, y))|x=y=1 = 594n2 − 682n + 242.

2. M2(D1(n)) = DxDy (f (x, y))|x = y = 1 = 1089n2 − 1357n + 501.

3. $\begin{array}{}{}^{m}{M}_{2}\left({D}_{1}\left(n\right)\right)={S}_{x}{S}_{y}\left(f\left(x,y\right)\right){|}_{x=y=1}=\frac{25}{4}{n}^{2}-\frac{151}{36}n+\frac{41}{36}.\end{array}$

4. $\begin{array}{}{R}_{\text{a}}\left({D}_{1}\left(n\right)\right)={D}_{x}^{\text{a}}{D}_{y}^{\text{a}}\left(f\left(x,y\right)\right){|}_{x=y=1}={2}^{2\text{a}+2}n+{6}^{\text{a}}\left(4n-4\right)+{8}^{\text{a}}\left(28n-16\right)+:\\ +{3}^{2\text{a}}\left(9{n}^{2}-13n+5\right)+{12}^{\text{a}}\left(36{n}^{2}-56n+24\right)+{2}^{4\text{a}}\left(36{n}^{2}-52n+20\right).\end{array}$

5. $\begin{array}{}R{R}_{\text{a}}\left({D}_{1}\left(n\right)\right)={S}_{x}^{\text{a}}{S}_{y}^{\text{a}}\left(f\left(x,y\right)\right){|}_{x=y=1}=\frac{1}{{2}^{2\text{a}-2}}n\frac{1}{{6}^{\text{a}}}\left(4n-4\right)+\frac{1}{{2}^{3\text{a}}}\left(28n-16\right)+\\ \frac{1}{{3}^{2\text{a}}}\left(9{n}^{2}-13n+5\right)+\frac{1}{{12}^{\text{a}}}\left(36{n}^{2}-56n+20\right)+\frac{1}{{2}^{4\text{a}}}\left(36{n}^{2}-52n+20\right).\end{array}$

6. $\begin{array}{}SSD\left({D}_{1}\left(n\right)\right)=\left({S}_{y}{D}_{x}+{S}_{x}{D}_{y}\right)\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}\left(f\left(x,y\right)\right)\phantom{\rule{negativethinmathspace}{0ex}}\phantom{\rule{negativethinmathspace}{0ex}}{|}_{x=y=1}=\\ =165{n}^{2}-160n+\frac{154}{3}.\end{array}$

7. $\begin{array}{}H\left({D}_{1}\left(n\right)\right)=2{S}_{x}J\left(f\left(x,y\right)\right){|}_{x=1}=\frac{156}{7}{n}^{2}-\frac{102}{5}n+\frac{692}{105}.\end{array}$

8. $\begin{array}{}I\left({D}_{1}\left(n\right)\right)={S}_{x}J{D}_{x}{D}_{y}\left(f\left(x,y\right){\right)}_{x=1}=\frac{2061}{14}{n}^{2}+\frac{5201}{30}n+\frac{13127}{210}.\end{array}$

9. $\begin{array}{}A\left({D}_{1}\left(n\right)\right)={S}_{x}{}^{3}{Q}_{-2}J{D}_{x}{}^{3}{D}_{y}{}^{3}\left(f\left(x,y\right){\right)}_{x=1}\\ =\frac{30788311}{24000}{n}^{2}-\frac{1858527979}{216000}n+\frac{130461491}{216000}.\end{array}$

## 2.2 M-polynomial and topological indices of Dominating David Derived network of second type

Let D2 (n) be the dominating David derived network of the second type. From Figure 2, we infer the following vertex and edge partition (Table 4, 5)

Table 4

The vertex partition of D2(n).

Table 5

Edge partition of D2(n).

#### Theorem 3

Let D2 (n) be the dominating David derived network of the second type. Then the M-Polynomial of D2 (n) is $M((D2(n);x,y)=4nx2y2+(18n2−22n+6)x2y3+(28n−16)x2y4+(36n2−56n+24)x3y4+(36n2−52n+20)x4y4.$

#### Proof

Let D2 (n) is the dominating David derived network of second type. It is easy to see form Figure 2 that there are three type of vertices in the vertex set of D2 (n): $V1(D2(n))={u∈V(D2(n)):du=2},V2(D2(n))={u∈V(D2(n)):du=3},V3(D2(n))={u∈V(D2(n)):du=4},$

with $|V1(D2(n))|=9n2+7n−5|V2(D2(n))|=18n2−26n+10$

and $|V3(D2(n))|=27n2−33n+12$

Also the edge set of D2 (n) has five type of edges: $E1(D2(n))={e=uv∈E(D2(n)):du=2,dv=2},E2(D2(n))={e=uv∈E(D2(n)):du=2,dv=3},E3(D2(n))={e=uv∈E(D2(n)):du=2,dv=4},E4(D2(n))={e=uv∈E(D2(n)):du=3,dv=4},E5(D2(n))={e=uv∈E(D2(n)):du=4,dv=4}$

such that |E1 (D2 (n))| = 4n, |E2 (D2 (n))| = 18n2 − 22n + 6, |E3(D2(n)) = 28n − 16, |E4(D2(n))| = 36n2 − 56n + 24, and |E5 (D2(n))| = 36n2 − 52n + 20.

In light of the definition of the M-polynomial, it is deduced that $M((D2(n));x,y)=∑i≤jmijxiyj=∑uv∈E1(D2(n))m22x2y2∑uv∈E2(D2(n))m23x2y3+∑uv∈E3(D2(n))m24x2y4+∑uv∈E4(D2(n))m34x3y4+∑uv∈E5(D2(n))m44x4y4=|E1(D2(n))|x2y2+|E2(D2(n))|x2y3+|E3(D2(n))|x2y4+|E4(D2(n))|x3y4+|E5(D2(n))|x4y4=4nx2y3+(18n2−22n+6)x2y3+(28n−16)x2y4+(36n2−−56n+24)x3y4+(36n2−52n+20)x4y4.$

Figure 14 is plotted using Maple 15. This suggests that values obtained by M-polynomial show different behaviors corresponding to different parameters x and y. We can control these values through these parameters.

Figure 5

Plot of M1(D1(n)).

Figure 6

Plot of M2(D1(n)).

Figure 7

Plot of mM2(D1(n)).

Figure 8

Plot of $\begin{array}{}{R}_{\frac{1}{2}}\end{array}$ (D1(n)).

Figure 9

Plot of RRα(D1(n)).

Figure 10

Plot of SSD (D1(n)).

Figure 11

Plot of H (D1(n)).

Figure 12

Plot of I (D1(n)).

Figure 13

Plot of A (D1(n)).

Figure 14

Plot of M-Polynomial of Dominating David Derived Network of second Type.

Now some degree-based topologcal indices of the dominating David derived network of the second type are computed from this M-polynomial.

#### Proposition 4

Let D2 (n) be the dominating David derived network of the second type

1. M1(D2(n)) = 630n2 − 734n + 262.

2. M2 (D2 (n)) = 1116n2 − 1396n + 516.

3. $\begin{array}{}{}^{m}{M}_{2}\left({D}_{2}\left(n\right)\right)=\frac{33}{4}{n}^{2}-\frac{85}{12}n+\frac{9}{4}.\end{array}$

4. Rα(D2 (n)) = (6α18 + 12α36 +124α36)n2 + (22α+2 − 6α22 + 23α28 − 12α56 − 24α52 + 6α6 − 23α16 +12α24 + 24α 20.

5. $\begin{array}{}R{R}_{\alpha }\left({D}_{2}\left(n\right)\right)=\frac{1}{{2}^{2\alpha }}4n+\frac{1}{{6}^{\alpha }}\left(18{n}^{2}-22n+6\right)+\frac{1}{{2}^{3\alpha }}\left(28n-16\right)+\\ +\frac{1}{{12}^{\alpha }}\left(36{n}^{2}-56n+24\right)+\frac{1}{{2}^{4\text{a}}}\left(36{n}^{2}-52n+20\right)\phantom{\rule{1em}{0ex}}.\end{array}$

6. $\begin{array}{}SSD\left({D}_{2}\left(n\right)\right)=186{n}^{2}-\frac{1285}{6}n+\frac{254}{3}.\end{array}$

7. $\begin{array}{}H\left({D}_{2}\left(n\right)\right)=\frac{927}{35}{n}^{2}-\frac{412}{15}n+\frac{1147}{105}.\end{array}$

8. $\begin{array}{}I\left({D}_{2}\left(n\right)\right)=\frac{5436}{35}{n}^{2}-\frac{2776}{15}n+\frac{7036}{105}.\end{array}$

9. $\begin{array}{}A\left({D}_{2}\left(n\right)\right)=\frac{496624}{375}{n}^{2}-\frac{5670736}{3375}n+\frac{2129744}{3375}.\end{array}$

#### Proof

Let M (D2(n); x, y) = f (x, y) = 4nx2 y3 + (18n2 − 22n + 6) x2 y3 + + (28n − 16) x2 y4 +(36n2 − 56n + 24) x3 y4 + (36n2 − 52n + 20) x4 y4.

Then, this yields $Dx(f(x,y))=8nx2y2+2(18n2−22n+6)x2y3+2(28n−16)x2y4+3(36n2−52n+24)x3y4+4(36n2−52n+20)x4y4,Dy(f(x,y))=8nx2y2+3(18n2−22n+6)x2y3+4(28n−16)x2y4+4(36n2−52n+24)x3y4+4(36n2−52n+20)x4y4,(DxDy)(f(x,y))=16nx2y2+6(18n2−22n+6)x2y3+8(28n−16)x2y4+12(36n2−52n+24)x3y4+16(36n2−52n+20)x4y4,SxSy(f(x,y))=nx2y2+16(18n2−22n+6)x2y3+18(28n−16)x2y4+112(36n2−52n+24)x3y4+116(36n2−52n+20)x4y4,DxαDyα(f(x,y))=22α4nx2y2+6α(18n2−22n+6)x2y3+8α(28n−16)x2y4+12α(36n2−52n+20)x4y4+42α(36n2−52n+24)x4y4,SxαSyα(f(x,y))=122α4nx2y2+16α(18n2−22n+6)x2y3+18α(28n−16)x2y4+112α(36n2−52n+24)x3y4+142α(36n2−52n+20)x4y4,DxSy(f(x,y))=4nx2y2+23(18n2−22n+6)x2y3+12(28n−16)x2y4+34(36n2−56n+24)x3y4+(36n2−52n+20)x4y4,SxDy(f(x,y))=4nx2y2+32(18n2−22n+6)x2y3+2(28n−16)x2y4+43(36n2−56n+24)x3y4+(36n2−52n+20)x4y4,SxJf(x,y)=nx4+15(18n2−22n+6)x5+16(28n−16)x6+17(36n2−56n+24)x7+18(36n2−52n+20)x8,SxJDxDy(f(x,y))=4nx4+65(18n2−22n+6)x5+43(28n−16)x6+127(36n2−56n+24)x7+2(36n2−52n+20)x8,Sx3Q−2JDx3Dy3f(x,y)=32m2+8(18n2−22n+6)x3+8(28n−16)x4+1728125(36n2−56n+24)x5+51227(36n2−52n+20)x6,$

Now from table 1

1. M1(D2(n)) = (Dx + Dy)(f(x, y))|x=y=1 = = 630n2 − 734n + 262.

2. M2(D2(n)) = DxDy(f(x, y))|x = y = 1 = 1116n2 − 1396n + 516.

3. $\begin{array}{}{}^{m}{M}_{2}\left({D}_{2}\left(n\right)\right)={S}_{x}{S}_{y}\left(f\left(x,y\right)\right){|}_{x=y=1}=\frac{33}{4}{n}^{2}-\frac{85}{12}n+\frac{9}{4}.\end{array}$

4. $\begin{array}{}{R}_{\alpha }\left({D}_{2}\left(n\right)\right)={D}_{x}^{\alpha }{D}_{y}^{\alpha }\left(f\left(x,y\right)\right){|}_{x=y=1}=\left({6}^{\alpha }18-{12}^{\alpha }36-\\ -{12}^{4\alpha }36\right){n}^{2}+\left({2}^{2\alpha +2}-{6}^{\alpha }22+{2}^{3\alpha }28\\ -{12}^{\alpha }56-{2}^{4\alpha }52\right)n+{6}^{\alpha }6-{2}^{3\alpha }16+{12}^{\alpha }24+{2}^{4\alpha }20.\end{array}$

5. $\begin{array}{}R{R}_{\alpha }\left({D}_{2}\left(n\right)\right)={S}_{x}^{\alpha }{S}_{y}^{\alpha }\left(f\left(x,y\right)\right){|}_{x=y=1}=\\ =\frac{1}{{2}^{\alpha }}4n+\frac{1}{{6}^{\alpha }}\left(18{n}^{2}-22n+6\right)+\frac{1}{{2}^{3\alpha }}\left(28n-16\right)\\ +\frac{1}{{12}^{\alpha }}\left(36{n}^{2}-56n+24\right)+\frac{1}{{2}^{4\alpha }}\left(36{n}^{2}-52n+20\right)\end{array}$

6. $\begin{array}{}SSD\left({D}_{2}\left(n\right)\right)=\left({S}_{y}{D}_{X}+{S}_{X}{D}_{y}\right)\phantom{\rule{negativethinmathspace}{0ex}}\left(f\left(x,y\right)\right){|}_{x=y=1}=\\ =186{n}^{2}-\frac{1285}{6}n+\frac{254}{3}\end{array}$

7. $\begin{array}{}H\left({D}_{2}\left(n\right)\right)=2{S}_{x}J\left(f\left(x,y\right)\right){|}_{x=1}=\frac{927}{35}{n}^{2}-\frac{412}{15}n+\frac{1147}{105}.\end{array}$

8. $\begin{array}{}I\left({D}_{2}\left(n\right)\right)={S}_{x}J{D}_{x}{D}_{y}\left(f\left(x,y\right){\right)}_{x=1}=\frac{5436}{35}{n}^{2}-\frac{2776}{15}n+\frac{7036}{105}\end{array}$

9. $\begin{array}{}A\left({D}_{2}\left(n\right)\right)={S}_{x}^{3}{Q}_{-2}J{D}_{x}{}^{3}{D}_{y}{}^{3}\left(f\left(x,y\right)\right){|}_{x=1}=\frac{496624}{375}{n}^{2}-\\ -\frac{5670736}{3375}n+\frac{2129744}{3375}\end{array}$

## 2.3 M-polynomial and topological indices of Dominating David Derived networks of type three

Let G = D1(n) be the dominating David derived network of third type. By means of Figure 3, it is given that:

#### Theorem 5

Let D3 (n) be the dominating David derived network of the third type. Then the M-Polynomial of D3 (n) is

$M(D3(n);x,y)=4nx2y2+(36n2−20n)x2y4+(72n2−108n+44)x4y4.$

#### Proof

Let D3(n) is the dominating David derived network of the third type. In view of Table 6, the vertex set of D3 (n) has two partitions:

$V1(D3(n))={u∈V(D3(n)):du=2},V2(D3(n))={u∈V(D3(n)):du=4},$

such that |V1(D3(n))| = 18n2 − 6n and |V2(D3(+))| = 45n2 − 59n + 22.

Table 6

The vertex partition of D3 (n).

Using Table 7, the edge set of D3 (n) has three partitions:

$E1(D3(n))={e=uv∈E(D3(n)):du=2,dv=2},E2(D3(n))={e=uv∈E(D3(n)):du=2,dv=4},E3(D3(n))={e=uv∈E(D3(n)):du=4,dv=4},$

which satisfy |E1 (D3 (n))| = 4n, |E2(D3(n))| = 36n2 − 20n and |E3(D3(n))| = 72n2 − 108n + 44.

Table 7

Edge partition of D3 (n).

Followed from the definition of the M-polynomial, we get

$M(D3(n);x,y)=∑i≤jmijxiyj=∑uv∈E1(D3(n))m22x2y2+∑uv∈E2(D3(n))m24x2y4+∑uv∈E3(D3(n))m44x4y4=|E1(D1(n))|x2y2+|E2(D3(n))|x2y4+|E3(D3(n))|x4y4=4nx2y2+(36n2−20n)x2y4+(72n2−108n+44)x4y4.$

Figure 24 is plotted by using Maple 15. This suggests that values obtained by M-polynomial show different behaviors corresponding to different parameters x and y. We can control values of M-polynomial through these parameters.

Figure 15

Plot of M1 (D2(n)).

Figure 16

Plot of M2 (D2(n)).

Figure 17

Plot of mM2 (D2(n)).

Figure 18

Plot of R-1/2(D2(n)).

Figure 19

Plot of RR-1/2(D2(n)).

Figure 20

Plot of SSD (D2(n)).

Figure 21

Plot of H D2(n)).

Figure 22

Plot of I D2(n)).

Figure 23

Plot of A D2(n)).

Figure 24

Plot of M-Polynomial of Dominating David Derived Network of third Type.

Now some degree-based topologcal indices of Dominating David Derived Network of the third type are computed from this M-polynomial.

#### Proposition 6

Let D3(n) be the Dominating David Derived Network of the third type

1. M1(D3(n)) = 792n2 − 968n + 352.

2. M2(D3(n)) = 1440n2 − 1872n + 704.

3. $\begin{array}{}{m}_{{M}_{2}\left({D}_{3}\left(n\right)\right)}=9{n}^{2}-\frac{33}{4}n+\frac{11}{4}.\end{array}$

4. Rα(D3(n)) = 9(23α+2 + 24α+3)n2 + (22α+2 − 5⋅3a+2 + + 27 ⋅ 24α+2)n + 11⋅ 24α+2.

5. $\begin{array}{}R{R}_{\alpha }\left({D}_{3}\left(n\right)\right)=\left(\frac{1}{{2}^{4\alpha -3}}+\frac{1}{{2}^{3\alpha -2}}\right)9{n}^{2}+\left(\frac{1}{{2}^{2\alpha -2}}-\frac{5}{{2}^{3\alpha -2}}-\\ -\frac{27}{{2}^{4\alpha -2}}\right)n+\frac{11}{{2}^{4\alpha -2}}.\end{array}$

6. SDD(D3(n)) = 234n2 − 258n + 88.

7. H(D3(n)) = 30n2 $\begin{array}{}-\frac{95}{3}\end{array}$ n + 11.

8. I (D3 (n)) = 192n2 $\begin{array}{}-\frac{716}{3}\end{array}$n + 88.

9. A(D3(n)) $\begin{array}{}=\frac{4960}{3}{n}^{2}-2176n+\frac{22528}{27}.\end{array}$

#### Proof

Let

$M(D3(n);x,y)=f(x,y)=4nx2y2+(36n2−20n)x2y4++(72n2−108n+44)x4y4$

Then, derived from this

$Dx(f(x,y))=8nx2y2+2(36n2−20n)x2y4+4(72n2−−108n+44)x4y4Dy(f(x,y))=8nx2y2+4(36n2−20n)x2y4+4(72n2−−108n+44)x4y4(DxDy)(f(x,y))=16nx2y2+8(36n2−20n)x2y4+16(72n2−−108n+44)x4y4SxSy(f(x,y))=nx2y2+18(36n2−20n)x2y4+116(72n2−−108n+44)x4y4.DxαDyα(f(x,y))=22α+2nx2y2+23α(36n2−20n)x2y4++24α(72n2−108n+44)x4y4.SxαSyα(f(x,y))=122α−2nx2y2123α+(36n2−20n)x2y4++124α(72n2−108n+44)x4y4.DxSy(f(x,y))=4nx2y2+12(36n2−20n)x2y4+(72n2−−108n+44)x4y4.SxDy(f(x,y))=4nx2y2+2(36n2−20n)x2y4+(72n2−−108n+44)x4y4.SxJf(x,y)=nx4+16(36n2−20n)x6+18(72n2−108n+44)x8.SxJDxDy(f(x,y))=4nx4+43(36n2−20n)x6+2(72n2−−108n+44)x8.SX3Q−2JDx3Dy3f(x,y)=25623nx2+51243(36n2−20n)x4++409663(72n2−108n+44)x6.$

Now from Table 1

1. Ml(D3(n)) = (Dx + Dy)(f(x, y))|x=y=1 = 792n2 − 968n + 352.

2. M2(D3(n)) = DxDy(f(x,y))|x=y=1 =1728n2 − 2032n + 704.

3. $\begin{array}{}{m}_{{M}_{2}}\left({D}_{3}\left(n\right)\right)={S}_{X}{S}_{y}\left(f\left(x,y\right)\right){|}_{x=y=1}=9{n}^{2}-\frac{33}{4}n+\frac{11}{4}\end{array}$

4. $\begin{array}{}{R}_{\alpha }\left({D}_{3}\left(n\right)\right)={D}_{x}^{\alpha }{D}_{y}^{\alpha }\left(f\left(x,y\right)\right){|}_{x=y=1}=9\left({2}^{3\alpha +2}+\\ +{2}^{4\text{a}+3}\right){n}^{2}+\left({2}^{2\text{a}+2}-5\cdot {3}^{\text{a}+2}+27\cdot {2}^{4\text{a}+2}\right)n+11\cdot {2}^{4\text{a}+2}.\end{array}$

5. $\begin{array}{}R{R}_{\mathrm{a}}\left({D}_{3}\left(n\right)\right)={S}_{x}^{\text{a}}{S}_{y}^{\text{a}}\left(f\left(x,y\right)\right){|}_{x=y=1}=\\ =\left(\frac{1}{{2}^{4\text{a}-3}}+\frac{1}{{2}^{3\text{a}-2}}\right)9{n}^{2}+\left(\frac{1}{{2}^{2\text{a}-2}}-\frac{5}{{2}^{3\text{a}-2}}-\frac{27}{{2}^{4\text{a}-2}}\right)n+\frac{11}{{2}^{4\text{a}-2}}.\end{array}$

6. SDD(D3 (n)) = (Sy Dx + Sx Dy )(f(x, y))|x=y=1 =

= 234n2 − 258n + 88.

7. $\begin{array}{}H\left({D}_{3}\left(n\right)\right)=2{S}_{X}J\left(f\left(x,\phantom{\rule{1em}{0ex}}y\right)\right){|}_{x=1}=30{n}^{2}-\frac{95}{3}n+11.\end{array}$

8. $\begin{array}{}I\left({D}_{3}\left(n\right)\right)={S}_{x}J{D}_{x}{D}_{y}\left(f\left(x,y\right){\right)}_{x=1}=192{n}^{2}-\frac{716}{3}n+88.\end{array}$

9. $\begin{array}{}A\left({D}_{3}\left(n\right)\right)={S}_{x}^{3}{Q}_{-2}J{D}_{x}^{3}{D}_{y}^{3}\left(f\left(x,y\right)\right){|}_{x=1}=\frac{4960}{3}{n}^{2}-\\ -2176n+\frac{22528}{27}.\end{array}$

Figure 25

Plot of M1(D3(n)).

Figure 26

Plot of M2(D3(n)).

Figure 27

Plot of mM2(D3(n)).

Figure 28

Plot of R-1/2(D3(n)).

Figure 29

Plot of RR-1/2(D3(n)).

Figure 30

Plot of SDD(D3(n)).

Figure 31

Plot of H(D3(n)).

Figure 32

Plot of I(D3(n)).

Figure 33

Plot of A(D3(n)).

## 3 Conclusion

M-polynomials of dominating David Derived networks of the first, second and third type were computed. Many degree-based topological indices of these networks form have been recovered from their M-polynomials. Note that first Zagreb index and some particular cases of Randic index was calculated directly in [1].

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Accepted: 2018-01-27

Published Online: 2018-03-20

Conflict of interest: Authors state no conflict of interest

Citation Information: Open Chemistry, Volume 16, Issue 1, Pages 201–213, ISSN (Online) 2391-5420,

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