M-Polynomials and Topological Indices of Dominating David Derived Networks

Shin Min Kang; Waqas Nazeer; Wei Gao; Deeba Afzal; Syeda Nausheen Gillani

doi:10.1515/chem-2018-0023

Open Access Published by De Gruyter Open Access March 20, 2018

M-Polynomials and Topological Indices of Dominating David Derived Networks

Shin Min Kang , Waqas Nazeer , Wei Gao , Deeba Afzal and Syeda Nausheen Gillani

From the journal Open Chemistry

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.

Keywords: degree-based topological index; Zagreb index; general Randić index; symmetric division index; M-polynomial; Networks

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.

Topological Index	Derivation from M (G; x, y)
First Zagreb	(D_x + D_y )(M (G; x, y)_x=y=1
Second Zagreb	(D_xD_y)(M (G; x, y))_x=y=1
Second Modified Zagreb	(S_xS_y )(M (G; x, y))_{_x=y=1}
Inverse Randić	(DxαDyα)(M (G; x, y))_{_x=y=1}
General Randić	(SxαSyα)(M (G; x, y ))_{_x=y=1}
Symmetric Division Index	(D_xS_y + S_xD_y)(M(G; x, y))_{_x=y=1}
Harmonic Index	2S_xJ(M (G; x, y))_x=1
Inverse sum Index	S_xJD_xD_y(M (G; x, y))_x=1
Augmented Zagreb Index	S_x³Q₋₂JD_x³D_y³ (G; x, y))

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 Main Results

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).

dv	2	3	4
Number of vertices	20n − 10	18n² − 26n +10	27n² − 33n +12

Table 3

The Edge partition of.

d_u	Number of edges
(2,2)	4n
(2,3)	4n − 4
(2,4)	28n − 6
(3,3)	9n² − 13n + 5
(3,4)	36n² − 56n + 24
(4,4)	36n² − 52n + 20

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)x⁴y⁴

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

M₁(D₁(n)) = 594n² − 682n + 242
M₂(D₁(n)) = 1089n² − 1357n + 501.
mM2(D1(n))=254n2−15136n+4136.
R_α (D₁(n)) = 2^{2 α} 4n + 6^α (4n − 4) + 8^α (28n − 16) + 3^2α (9n² − 13n + 5) + + 12^α (36n² − 56n + 24) +4^2α (36n² − 52n + 20).
RRα(D1(n))=122α−2n+16α(4n−4)+123α(28n−16)+132α(9n2−13n+5112α(36n2−56n+20)+124α(36n2−52n+20).
SSD(D₁(n)) = 165n² − 160n + 1543.
H(D1(n))=1567n2−1025n+692105.
I(D1(n))=206114n2+520130n+13127210.
A(D1(n))=3078831124000n2−1858527979216000n+130461491216000.

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:

M₁(D₁(n)) = (D_x + D_y)(f (x, y))|_x=y=1 = 594n² − 682n + 242.
M₂(D₁(n)) = D_xD_y (f (x, y))|_{x = y = 1} = 1089n² − 1357n + 501.
mM2(D1(n))=SxSy(f(x,y))|x=y=1=254n2−15136n+4136.
Ra(D1(n))=DxaDya(f(x,y))|x=y=1=22a+2n+6a(4n−4)+8a(28n−16)+:+32a(9n2−13n+5)+12a(36n2−56n+24)+24a(36n2−52n+20).
RRa(D1(n))=SxaSya(f(x,y))|x=y=1=122a−2n16a(4n−4)+123a(28n−16)+132a(9n2−13n+5)+112a(36n2−56n+20)+124a(36n2−52n+20).
SSD(D1(n))=(SyDx+SxDy)(f(x,y))|x=y=1==165n2−160n+1543.
H(D1(n))=2SxJ(f(x,y))|x=1=1567n2−1025n+692105.
I(D1(n))=SxJDxDy(f(x,y))x=1=206114n2+520130n+13127210.
A(D1(n))=Sx3Q−2JDx3Dy3(f(x,y))x=1=3078831124000n2−1858527979216000n+130461491216000.

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).

d_v	2	3	4
Number of vertices	9 n² + 7 n − 5	18n² − 26n +10	27n² − 33n+ 12

Table 5

Edge partition of D₂(n).

d_u	Number of edges
(2,2)	4n
(2,3)	18n² − 22n + 6
(2,4)	28n − 16
(3,4)	36 n² − 56n+24
(4,4)	36n² − 52n + 20

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)).

$Figure 8 Plot of R12$\begin{array}{} R_{\frac12} \end{array} $ (D1(n)).$

Figure 8

Plot of R12 (D₁(n)).

Figure 9

Plot of RR_α(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

M₁(D₂(n)) = 630n² − 734n + 262.
M₂ (D₂ (n)) = 1116n² − 1396n + 516.
mM2(D2(n))=334n2−8512n+94.
R_α(D₂ (n)) = (6^α18 + 12^α36 +12^4α36)n² + (2^2α+2 − 6^α22 + 2^3α28 − 12^α56 − 2^4α52 + 6^α6 − 2^3α16 +12^α24 + 2^4α 20.
RRα(D2(n))=122α4n+16α(18n2−22n+6)+123α(28n−16)++112α(36n2−56n+24)+124a(36n2−52n+20).
SSD(D2(n))=186n2−12856n+2543.
H(D2(n))=92735n2−41215n+1147105.
I(D2(n))=543635n2−277615n+7036105.
A(D2(n))=496624375n2−56707363375n+21297443375.

Proof

Let M (D₂(n); x, y) = f (x, y) = 4nx²y³ + (18n² − 22n + 6) x²y³ + + (28n − 16) x²y⁴ +(36n² − 56n + 24) x³y⁴ + (36n² − 52n + 20) x⁴y⁴.

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

M₁(D₂(n)) = (D_x + D_y)(f(x, y))|_x=y=1 = = 630n² − 734n + 262.
M₂(D₂(n)) = D_xD_y(f(x, y))|_{x = y = 1} = 1116n² − 1396n + 516.
mM2(D2(n))=SxSy(f(x,y))|x=y=1=334n2−8512n+94.
Rα(D2(n))=DxαDyα(f(x,y))|x=y=1=(6α18−12α36−−124α36)n2+(22α+2−6α22+23α28−12α56−24α52)n+6α6−23α16+12α24+24α20.
RRα(D2(n))=SxαSyα(f(x,y))|x=y=1==12α4n+16α(18n2−22n+6)+123α(28n−16)+112α(36n2−56n+24)+124α(36n2−52n+20)
SSD(D2(n))=(SyDX+SXDy)(f(x,y))|x=y=1==186n2−12856n+2543
H(D2(n))=2SxJ(f(x,y))|x=1=92735n2−41215n+1147105.
I(D2(n))=SxJDxDy(f(x,y))x=1=543635n2−277615n+7036105
A(D2(n))=Sx3Q−2JDx3Dy3(f(x,y))|x=1=496624375n2−−56707363375n+21297443375

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).

d_v	2	4
Number of vertices	18n² − 6n	45n² − 59n + 22

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).

d_u	Number of edges
(2,2)	4n
(2,4)	36n² − 20n
(4,4)	72n² − 108n + 44

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

M₁(D₃(n)) = 792n² − 968n + 352.
M₂(D₃(n)) = 1440n² − 1872n + 704.
mM2(D3(n))=9n2−334n+114.
R_α(D₃(n)) = 9(2^3α+² + 2^4α+³)n² + (2^2α+² − 5⋅3^a+² + + 27 ⋅ 2^4α+²)n + 11⋅ 2^4α+².
RRα(D3(n))=(124α−3+123α−2)9n2+(122α−2−523α−2−−2724α−2)n+1124α−2.
SDD(D₃(n)) = 234n² − 258n + 88.
H(D₃(n)) = 30n²−953n + 11.
I (D₃ (n)) = 192n²−7163n + 88.
A(D₃(n)) =49603n2−2176n+2252827.

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

M_l(D₃(n)) = (D_x + D_y)(f(x, y))|_x=y=1 = 792n² − 968n + 352.
M₂(D₃(n)) = D_xD_y(f(x,y))|_x=y=1 =1728n² − 2032n + 704.
mM2(D3(n))=SXSy(f(x,y))|x=y=1=9n2−334n+114
Rα(D3(n))=DxαDyα(f(x,y))|x=y=1=9(23α+2++24a+3)n2+(22a+2−5⋅3a+2+27⋅24a+2)n+11⋅24a+2.
RRa(D3(n))=SxaSya(f(x,y))|x=y=1==(124a−3+123a−2)9n2+(122a−2−523a−2−2724a−2)n+1124a−2.
SDD(D3 (n)) = (S_yD_x + S_xD_y )(f(x, y))|_x=y=1 =
= 234n² − 258n + 88.
H(D3(n))=2SXJ(f(x,y))|x=1=30n2−953n+11.
I(D3(n))=SxJDxDy(f(x,y))x=1=192n2−7163n+88.
A(D3(n))=Sx3Q−2JDx3Dy3(f(x,y))|x=1=49603n2−−2176n+2252827.

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].

Conflict of interest: Authors state no conflict of interest

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Received: 2017-08-21

Accepted: 2018-01-27

Published Online: 2018-03-20

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

M-Polynomials and Topological Indices of Dominating David Derived Networks

Abstract

1 Introduction

2 Main Results

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

Theorem 1

Proof

Proposition 2

Proof

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

Theorem 3

Proof

Proposition 4

Proof

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

Theorem 5

Proof

Proposition 6

Proof

3 Conclusion

Reference

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