# 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

## 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.

 Topological Index Derivation from M (G; x, y) First Zagreb (Dx + Dy )(M (G; x, y)x=y=1 Second Zagreb (DxDy)(M (G; x, y))x=y=1 Second Modified Zagreb (SxSy )(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 (DxSy + SxDy)(M(G; x, y))x=y=1 Harmonic Index 2SxJ(M (G; x, y))x=1 Inverse sum Index SxJDxDy(M (G; x, y))x=1 Augmented Zagreb Index Sx3Q−2JDx3Dy3 (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).

dv234
Number of vertices20n − 1018n2 − 26n +1027n2 − 33n +12

Table 3

The Edge partition of.

duNumber 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))={uV(D1(n)):du=2},V2(D1(n))={uV(D1(n)):du=3},V3(D1(n))={uV(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=uvE(D1(n)):du=2,dv=2},E2(D1(n))={e=uvE(D1(n)):du=2,dv=3},E3(D1(n))={e=uvE(D1(n)):du=2,dv=4},E4(D1(n))={e=uvE(D1(n)):du=3,dv=3},E5(D1(n))={e=uvE(D1(n)):du=3,dv=4},E6(D1(n))={e=uvE(D1(n)):du=4,dv=4}.

By means of Figure 1, this gives

|E1(D1(n))|=4n,|E2(D1(n))|=4n4,|E3(D1(n))|=28n16,|E4(D1(n))|=9n213n+5,|E5(D1(n))|=36n256n+24,|E6(D1(n))|=36n252n+20.

Now according to the definition of the M-polynomial, this gives

M(D1(n);x,y)=mijxjyj=uvE1(D1(n))m22x2y2ij+uvE2(D1(n))m23x2y3+uvE3(D1(n))m24x2y4+uvE4(D1(n))m33x3y3+uvE5(D1(n))m34x3y4+uvE6(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+(4n4)x2y3+(28n16)x2y4+(9n213n+5)x3y3+(36n256n+24)x3y4+(36n252n+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. mM2(D1(n))=254n215136n+4136.

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. RRα(D1(n))=122α2n+16α(4n4)+123α(28n16)+132α(9n213n+5112α(36n256n+20)+124α(36n252n+20).

6. SSD(D1(n)) = 165n2 − 160n + 1543.

7. H(D1(n))=1567n21025n+692105.

8. I(D1(n))=206114n2+520130n+13127210.

9. A(D1(n))=3078831124000n21858527979216000n+130461491216000.

### Proof

Let

f(x,y)=4nx2y2+(4n4)x2y3+(28n16)x2y4+(9n23n+5)x3y3+(36n256n+24)x3y4+(36n252n+20)x4y4.

Then

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. mM2(D1(n))=SxSy(f(x,y))|x=y=1=254n215136n+4136.

5. RRa(D1(n))=SxaSya(f(x,y))|x=y=1=122a2n16a(4n4)+123a(28n16)+132a(9n213n+5)+112a(36n256n+20)+124a(36n252n+20).

6. SSD(D1(n))=(SyDx+SxDy)(f(x,y))|x=y=1==165n2160n+1543.

7. H(D1(n))=2SxJ(f(x,y))|x=1=1567n21025n+692105.

8. I(D1(n))=SxJDxDy(f(x,y))x=1=206114n2+520130n+13127210.

9. A(D1(n))=Sx3Q2JDx3Dy3(f(x,y))x=1=3078831124000n21858527979216000n+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).

dv234
Number of vertices9 n2 + 7 n − 518n2 − 26n +1027n2 − 33n+ 12

Table 5

Edge partition of D2(n).

duNumber 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+(18n222n+6)x2y3+(28n16)x2y4+(36n256n+24)x3y4+(36n252n+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))={uV(D2(n)):du=2},V2(D2(n))={uV(D2(n)):du=3},V3(D2(n))={uV(D2(n)):du=4},

with

|V1(D2(n))|=9n2+7n5|V2(D2(n))|=18n226n+10

and

|V3(D2(n))|=27n233n+12

Also the edge set of D2 (n) has five type of edges:

E1(D2(n))={e=uvE(D2(n)):du=2,dv=2},E2(D2(n))={e=uvE(D2(n)):du=2,dv=3},E3(D2(n))={e=uvE(D2(n)):du=2,dv=4},E4(D2(n))={e=uvE(D2(n)):du=3,dv=4},E5(D2(n))={e=uvE(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)=ijmijxiyj=uvE1(D2(n))m22x2y2uvE2(D2(n))m23x2y3+uvE3(D2(n))m24x2y4+uvE4(D2(n))m34x3y4+uvE5(D2(n))m44x4y4=|E1(D2(n))|x2y2+|E2(D2(n))|x2y3+|E3(D2(n))|x2y4+|E4(D2(n))|x3y4+|E5(D2(n))|x4y4=4nx2y3+(18n222n+6)x2y3+(28n16)x2y4+(36n256n+24)x3y4+(36n252n+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 (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. mM2(D2(n))=334n28512n+94.

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

5. RRα(D2(n))=122α4n+16α(18n222n+6)+123α(28n16)++112α(36n256n+24)+124a(36n252n+20).

6. SSD(D2(n))=186n212856n+2543.

7. H(D2(n))=92735n241215n+1147105.

8. I(D2(n))=543635n2277615n+7036105.

9. A(D2(n))=496624375n256707363375n+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(18n222n+6)x2y3+2(28n16)x2y4+3(36n252n+24)x3y4+4(36n252n+20)x4y4,Dy(f(x,y))=8nx2y2+3(18n222n+6)x2y3+4(28n16)x2y4+4(36n252n+24)x3y4+4(36n252n+20)x4y4,(DxDy)(f(x,y))=16nx2y2+6(18n222n+6)x2y3+8(28n16)x2y4+12(36n252n+24)x3y4+16(36n252n+20)x4y4,SxSy(f(x,y))=nx2y2+16(18n222n+6)x2y3+18(28n16)x2y4+112(36n252n+24)x3y4+116(36n252n+20)x4y4,DxαDyα(f(x,y))=22α4nx2y2+6α(18n222n+6)x2y3+8α(28n16)x2y4+12α(36n252n+20)x4y4+42α(36n252n+24)x4y4,SxαSyα(f(x,y))=122α4nx2y2+16α(18n222n+6)x2y3+18α(28n16)x2y4+112α(36n252n+24)x3y4+142α(36n252n+20)x4y4,DxSy(f(x,y))=4nx2y2+23(18n222n+6)x2y3+12(28n16)x2y4+34(36n256n+24)x3y4+(36n252n+20)x4y4,SxDy(f(x,y))=4nx2y2+32(18n222n+6)x2y3+2(28n16)x2y4+43(36n256n+24)x3y4+(36n252n+20)x4y4,SxJf(x,y)=nx4+15(18n222n+6)x5+16(28n16)x6+17(36n256n+24)x7+18(36n252n+20)x8,SxJDxDy(f(x,y))=4nx4+65(18n222n+6)x5+43(28n16)x6+127(36n256n+24)x7+2(36n252n+20)x8,Sx3Q2JDx3Dy3f(x,y)=32m2+8(18n222n+6)x3+8(28n16)x4+1728125(36n256n+24)x5+51227(36n252n+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. mM2(D2(n))=SxSy(f(x,y))|x=y=1=334n28512n+94.

4. Rα(D2(n))=DxαDyα(f(x,y))|x=y=1=(6α1812α36124α36)n2+(22α+26α22+23α2812α5624α52)n+6α623α16+12α24+24α20.

5. RRα(D2(n))=SxαSyα(f(x,y))|x=y=1==12α4n+16α(18n222n+6)+123α(28n16)+112α(36n256n+24)+124α(36n252n+20)

6. SSD(D2(n))=(SyDX+SXDy)(f(x,y))|x=y=1==186n212856n+2543

7. H(D2(n))=2SxJ(f(x,y))|x=1=92735n241215n+1147105.

8. I(D2(n))=SxJDxDy(f(x,y))x=1=543635n2277615n+7036105

9. A(D2(n))=Sx3Q2JDx3Dy3(f(x,y))|x=1=496624375n256707363375n+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+(36n220n)x2y4+(72n2108n+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))={uV(D3(n)):du=2},V2(D3(n))={uV(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).

dv24
Number of vertices18n2 − 6n45n2 − 59n + 22

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

E1(D3(n))={e=uvE(D3(n)):du=2,dv=2},E2(D3(n))={e=uvE(D3(n)):du=2,dv=4},E3(D3(n))={e=uvE(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).

duNumber 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)=ijmijxiyj=uvE1(D3(n))m22x2y2+uvE2(D3(n))m24x2y4+uvE3(D3(n))m44x4y4=|E1(D1(n))|x2y2+|E2(D3(n))|x2y4+|E3(D3(n))|x4y4=4nx2y2+(36n220n)x2y4+(72n2108n+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. mM2(D3(n))=9n2334n+114.

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

5. RRα(D3(n))=(124α3+123α2)9n2+(122α2523α22724α2)n+1124α2.

6. SDD(D3(n)) = 234n2 − 258n + 88.

7. H(D3(n)) = 30n2953n + 11.

8. I (D3 (n)) = 192n27163n + 88.

9. A(D3(n)) =49603n22176n+2252827.

### Proof

Let

M(D3(n);x,y)=f(x,y)=4nx2y2+(36n220n)x2y4++(72n2108n+44)x4y4

Then, derived from this

Dx(f(x,y))=8nx2y2+2(36n220n)x2y4+4(72n2108n+44)x4y4Dy(f(x,y))=8nx2y2+4(36n220n)x2y4+4(72n2108n+44)x4y4(DxDy)(f(x,y))=16nx2y2+8(36n220n)x2y4+16(72n2108n+44)x4y4SxSy(f(x,y))=nx2y2+18(36n220n)x2y4+116(72n2108n+44)x4y4.DxαDyα(f(x,y))=22α+2nx2y2+23α(36n220n)x2y4++24α(72n2108n+44)x4y4.SxαSyα(f(x,y))=122α2nx2y2123α+(36n220n)x2y4++124α(72n2108n+44)x4y4.DxSy(f(x,y))=4nx2y2+12(36n220n)x2y4+(72n2108n+44)x4y4.SxDy(f(x,y))=4nx2y2+2(36n220n)x2y4+(72n2108n+44)x4y4.SxJf(x,y)=nx4+16(36n220n)x6+18(72n2108n+44)x8.SxJDxDy(f(x,y))=4nx4+43(36n220n)x6+2(72n2108n+44)x8.SX3Q2JDx3Dy3f(x,y)=25623nx2+51243(36n220n)x4++409663(72n2108n+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. mM2(D3(n))=SXSy(f(x,y))|x=y=1=9n2334n+114

4. Rα(D3(n))=DxαDyα(f(x,y))|x=y=1=9(23α+2++24a+3)n2+(22a+253a+2+2724a+2)n+1124a+2.

5. RRa(D3(n))=SxaSya(f(x,y))|x=y=1==(124a3+123a2)9n2+(122a2523a22724a2)n+1124a2.

6. SDD(D3 (n)) = (SyDx + SxDy )(f(x, y))|x=y=1 =

= 234n2 − 258n + 88.

7. H(D3(n))=2SXJ(f(x,y))|x=1=30n2953n+11.

8. I(D3(n))=SxJDxDy(f(x,y))x=1=192n27163n+88.

9. A(D3(n))=Sx3Q2JDx3Dy3(f(x,y))|x=1=49603n22176n+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].

1. Conflict of interest: Authors state no conflict of interest

## Reference

[1] Imran M., Baig A.Q., Ali H., On topological properties of dominating David derived networks. Canadian Journal of Chemistry, 2015, 94(2), 137-148.10.1139/cjc-2015-0185Search in Google Scholar

[2] Deutsch E., Klavzar S., M-Polynomial and degree-based topological indices. Iran. J. Math. Chem., 2015, 6, 93–102.Search in Google Scholar

[3] Mihalić Z., Trinajstić N., A graph-theoretical approach to structure-property relationships. CRC press, 1992.10.1021/ed069p701Search in Google Scholar

[4] Star of David [online]. Available from http.//Wikipedia.org/wiki/starofDavid.Search in Google Scholar

[5] Simonraj F., George A., GRAPH-HOC 2012. 4. 11. 10.5121/jgraphhoc.2012.402.Search in Google Scholar

[6] Munir M., Nazeer W., Rafique S., Kang S.M., M-polynomial and related topological indices of Nanostar dendrimers. Symmetry., 2016, 8(9), 97, 10.3390/sym8090097.10.3390/sym8090097Search in Google Scholar

[7] Munir M., Nazeer W., Rafique S., Nizami A.R., Kang S.M., M-polynomial and degree-based topological indices of titania nanotubes. Symmetry., 2016, 8(11), 117, 10.3390/sym8110117.10.3390/sym8110117Search in Google Scholar

[8] Munir M., Nazeer W., Rafique S., Kan S.M., M-Polynomial and Degree-Based Topological Indices of Polyhex Nanotubes. Symmetry., 2016, 8(12), 149, 10.3390/sym8120149.10.3390/sym8120149Search in Google Scholar

[9] Riaz M., Gao W., Baig A.Q., M-POLYNOMIALS AND DEGREEBASED TOPOLOGICAL INDICES OF SOME FAMILIES OF CONVEX POLYTOPES. Open J. Math. Sci., 2018, In Press.10.30538/oms2018.0014Search in Google Scholar

[10] Munir M., Nazeer W., Shahzadi S., Kang S.M., Some invariants of circulant graphs. Symmetry, 2016, 8(11), 134, 10.3390/sym8110134.10.3390/sym8110134Search in Google Scholar

[11] Kwun Y.C., Munir M., Nazeer W., Rafique S., Kang S.M., M-Polynomials and topological indices of V-Phenylenic Nanotubes and Nanotori. Scientific reports, 2017, 7, 8756.10.1038/s41598-017-08309-ySearch in Google Scholar PubMed PubMed Central

[12] Wiener H., Structural determination of paraffin boiling points. J. Am. Chem. Soc., 1947, 69, 17–20.10.1021/ja01193a005Search in Google Scholar PubMed

[13] Randić M., On characterization of molecular branching. J. Am. Chem. Soc., 1975, 97, 6609–6615.10.1021/ja00856a001Search in Google Scholar

[14] Bollobas B., Erdos P., Graphs of extremal weights. Ars. Combin., 1998, 50, 225–233.Search in Google Scholar

[15] Amic D., Beslo D., Lucic B., Nikolic S., Trinajstić N., The Vertex-Connectivity Index Revisited. J. Chem. Inf. Comput. Sci., 1998, 38, 819–822.10.1021/ci980039bSearch in Google Scholar

[16] Randić M., On History of the Randić Index and Emerging Hostility toward Chemical Graph Theory. MATCH Commun. Math. Comput. Chem., 2008, 59, 5-124.Search in Google Scholar

[17] Randić M., The Connectivity Index 25 Years After. J. Mol. Graphics Modell., 2001, 20, 19–35.10.1016/S1093-3263(01)00098-5Search in Google Scholar

[18] Huang Y., Liu B., Gan, L., Augmented Zagreb Index of Connected Graphs. MATCH Commun. Math. Comput. Chem., 2012, 67, 483-494.Search in Google Scholar

[19] Furtula B., Graovac A., Vukičević D., Augmented Zagreb index. J. Math. Chem., 2010, 48, 370–380.10.1007/s10910-010-9677-3Search in Google Scholar

[20] Gao W., Wang Y.Q., Basavanagoud B., and Jamil M.K., Characteristics studies of molecular structures in drugs. Saudi Pharmaceutical J., 2017, 25, 580-586.10.1016/j.jsps.2017.04.027Search in Google Scholar PubMed PubMed Central

[21] Gao W., Wang W.F., The eccentric connectivity polynomial of two classes of nanotubes. Chaos Soliton. Fract., 2016, 89, 290–294.10.1016/j.chaos.2015.11.035Search in Google Scholar

[22] Sardar M.S., Zafar, S., Farahani M.R., The Generalized Zagreb Index of Capra-Designed Planar Benzenoid Series $Ca_K(C_6)$, Open J. Math. Sci., 2017, 1, 44-51.10.30538/oms2017.0005Search in Google Scholar

[23] Matti ul Rehman H., Sardar, M.R., Raza, A., Computing Topological Indices of Hex Board and iIts Line Graph, Open J. Math. Sci., 2017, 1, 62-71.10.30538/oms2017.0007Search in Google Scholar

[24] Sardar M.S., Pan X.F., Gao W., Farahani M.R., Computing Sanskruti Index Of Titania Nanotubes, Open J. Math. Sci., 2017, 1, 126–131.10.30538/oms2017.0012Search in Google Scholar

[25] De N., Hyper Zagreb Index of Bridge and Chain Graphs, Open J. Math. Sci., 2018, 2(1), 1-17.10.30538/oms2018.0013Search in Google Scholar

[26] Gao W., Muzaffar B., Nazeer W., K-Banhatti and K-Hyper Banhatti Indices of Dominating David Derived Network, Open J. Math. Anal., 2017, 1, 13-24.10.30538/psrp-oma2017.0002Search in Google Scholar

[27] Siddiqui H., Farahani M. R., Forgotten Polynomial and Forgotten Index of Certain Interconnection Networks, Open J. Math. Anal., 2017, 1, 45-60.10.30538/psrp-oma2017.0005Search in Google Scholar