# Computing Topological Descriptors and Polynomials of Certain 2D Chemical Structures

Authors: Muhammad Mubashir Izhar; Zahida Perveen; Dalal Alrowaili; Mehran Azeem; Imran Siddique; Shahid Imran; Muhammad Shoaib Sardar
Journal: Journal of Chemistry
 (2021)

---

## Abstract

In the fields of mathematical chemistry, a topological index, also known as a connectivity index, is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are an analytical framework of a graph which portray its topology and are mostly equal graphs. Topological indices (TIs) are numeral quantities that are used to foresee the natural correlation among the physicochemical properties of the chemical compounds in their fundamental network. TIs show an essential role in the theoretical abstract and environmental chemistry and pharmacology. In this paper, we compute many latest developed degree-based TIs. An analogy among the computed different versions of the TIs with the help of the numerical values and their graphs is also included .In this article, we compute the first Zagreb index, second Zagreb index, hyper Zagreb index, ABC Index, GA Index, and first Zagreb polynomial and second Zagreb polynomial of chemical graphs polythiophene, nylon 6,6, and the backbone structure of DNA.


---

## Body

## 1. Introduction

In mathematics, the graph is an abstract structure formed by a set of end points called nodes or verticesV(G) which are joined by a set of lines called edges E(G). The order and size of a graph are the total number of vertices and total number of edges of the graph G, respectively. Every graph includes compounds of covalent bonds or molecules. Usually in graph theory, we combine graphs with chemistry to solve different chemical graphs. The branch of mathematical chemistry in which apparatus of the graph hypothesis are applied to display the substance numerically is called chemical graph theory. Topological index is the representation of a chemical graph that must be a structural invariant which collects the information about the number of elements and their connectivity which is used to model the chemical graph. Topological index and polynomials are the presentation of changeless chemical graphs in which we assemble complete statistics of these chemical graphs. Through this methodology, the number of elements, edges, vertices, and their bonding are determined easily. For the past 40 years, topological indices are being used to study the features of drug molecules. According to the drug examination, there is a strong relation between the properties of drug molecules pharmacodynamics.Weiner [1] was in search of the phenomenon of the effects of structural differences of chemical compounds on boiling points of these chemical compounds. In 1947, he wrote the first paper “structural determination of paraffin points,” where he satisfactory got results by this approach. In the last decade, the graph theory has found the maximum use in area of research of chemical graphs. Chemists have provided a variety of useful tools such as topological indices and polynomials. Cheminformatics is a new subject which is recently introduced that is the combination of mathematics, chemistry, and information technology which deals with structural properties of chemical graphs and their relationships which helps to predict bioactivity of the chemical compounds. In the QSAR/QSPR study, physiochemical properties and topological indices such as the 1st multiple Zagreb index, 2nd multiple Zagreb index, hyper Zagreb index, and Zagreb polynomials are used to predict the bioactivity of chemical compounds. Siddiqui et al. [2] put light on Zagreb indices and Zagreb polynomials of some nanostar dendrimers.The graph nylon 6,6 is a grid graph which is actually a chemical graph expanded horizontally top cycles and vertically to q cycles. C5 connected with a set of q − 1 cycles called polythiophene, and the third graph is the backbone structure of DNA. In this research, we will find the 1st and 2nd Zagreb index, first multiple Zagreb index, second multiple Zagreb index, hyper Zagreb index, and Zagreb polynomials of chemical graphs nylon 6,6, polythiophene, and the backbone structure of DNA.Gutman and Trinajstic [3] stated in their study that the first and second Zagreb indices are based on the following:(1)M1G=∑βγ∈EG∂β+∂γ,M2G=∑βγ∈EG∂β×∂γ.In 2013, According to Shirdel et al. [4], “hyper Zagreb indices” were defined as(2)HMG=∑βγ∈EG∂β+∂γ2.The most popular connectivity topological index is the “atom-bond connectivity (ABC)” index which is presented by Estrada et al. [5]. Let G be the graph, and the ABC index is illustrated as(3)ABCG=∑βγ∈EG∂β+∂γ−2∂β∂γ.Another most popular topological descriptor of connectivity is the “geometrically arithmetic (GA) index” that was presented by Vukicevic et al. [6]. We consider the graph G; then, the GA index is defined as(4)GAG=∑βγ∈EG2∂β×∂γ∂β+∂γ.Multiple Zagreb indices introduced by Ghorbani and Azimi [7] are defined as(5)PM1G=∏βγ∈EG∂β+∂γ,PM2G=∏βγ∈EG∂β×∂γ.Eliasi et al. [8] defined the first Zagreb polynomial M1G,x and second Zagreb polynomial M2G,x as(6)M1G,x=∑βγ∈EGx∂β+∂γ,M2G,x=∑βγ∈EGx∂β×∂γ.For details on the extensive research activity onHMG, PM1G, and PM2G indices and M1G,x and M2G,x polynomials, see [9–11]. Now, we compute all the abovementioned topological indices and polynomials of chemical graphs nylon 6,6, polythiophene, and the backbone structure of DNA.


## 2. Main Results

In this section, we will compute the main results of two-dimensional structures of nylon 6,6, polythiophene, and the backbone structure of DNA. This section is divided into three parts. In the first part, we compute the topological indices and polynomials of nylon 6,6, in the second part, we compute the topological indices and polynomials of polythiophene, and in the third part, we compute the topological indices and polynomials of the backbone structure of DNA.


### 2.1. Topological Indices and Polynomials of Nylon6,6

Theorem 1.
LetR1 be the graph nylon 6,6, as shown in Figure 1. Then,Figure 1
Nylon 6,6.(7)
M
1
R
1
=
46
p
q
+
28
p
+
12
q
−
6
,
M
2
R
1
=
53
p
q
+
27
p
+
11
q
−
11
,
HM
R
1
=
216
p
q
+
110
p
+
46
q
−
52
,
PM
1
R
1
=
4
5
p
q
+
6
p
+
2
q
+
7
×
5
4
p
q
+
2
p
+
2
q
−
8
×
6
p
q
−
p
−
q
+
1
,
PM
2
R
1
=
4
5
p
q
+
6
p
+
2
q
+
7
×
6
4
p
q
+
2
p
+
2
q
−
8
×
9
p
q
−
p
−
q
+
1
,
M
1
R
1
,
x
=
5
p
q
+
6
p
+
2
q
+
7
x
4
+
4
p
q
+
2
p
+
2
q
−
8
x
5
+
p
q
−
p
−
q
+
1
x
6
,
M
2
R
1
,
x
=
5
p
q
+
6
p
+
2
q
+
7
x
4
+
4
p
q
+
2
p
+
2
q
−
8
x
6
+
p
q
−
p
−
q
+
1
x
9
.Proof.
R
1 is isomorphic to nylon 6,6. Total edges of R1 are 10pq + 7p + 3q while there are horizontally 10p + 3 edges and vertically up to q with 7p edges at the end in addition. Edge partition is described in Table 1 for the chemical graph nylon 6,6 based on the degree sum of end vertices of each edge.(8)E2,2R1=β,γ∈ER1|∂β=2,∂γ=2,E2,3R1=β,γ∈ER1|∂β=2,∂γ=3,E3,3R1=β,γ∈ER1|∂β=3,∂γ=3.Table 1
Partition of the edges of N 6,6(p,q) with respect to the degrees of end nodes.
∂
β
,
∂
γ
E
∂
β
,
∂
γ
(2,2)
5pq + 6p + 2q + 7
(2,3)
4pq+2p+2q − 8
(3,3)
pq − p − q + 1


#### 2.1.1. First Zagreb Index

(9)
M
1
R
1
=
∑
β
γ
∈
E
R
1
∂
β
+
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
+
∑
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
+
∑
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
,
=
4
E
22
R
1
+
5
E
23
R
1
+
6
E
33
R
1
,
M
1
R
1
=
46
p
q
+
28
p
+
12
q
−
6.


#### 2.1.2. Second Zagreb Index

(10)
M
2
R
1
=
∑
β
γ
∈
E
R
1
∂
β
×
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
∂
β
×
∂
γ
+
∑
β
γ
∈
E
23
R
1
∂
β
×
∂
γ
+
∑
β
γ
∈
E
33
R
1
∂
β
×
∂
γ
,
=
4
E
22
R
1
+
6
E
23
R
1
+
9
E
33
R
1
,
M
2
R
1
=
53
p
q
+
27
p
+
11
q
−
11.


#### 2.1.3. Hyper Zagreb Index

(11)
HM
R
1
=
∑
β
γ
∈
E
R
1
∂
β
+
∂
γ
2
,
=
∑
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
2
+
∑
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
2
+
∑
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
2
,
=
4
2
E
22
R
1
+
5
2
E
23
R
1
+
6
2
E
33
R
1
,
=
216
p
q
+
110
p
+
46
q
−
52.In Table2, the relation of the 1st Zagreb index, 2nd Zagreb index, and hyper Zagreb index with one another is described.Table 2
Comparison ofM1, M2, and HM for N 6,6.
(p,q)
M
1
R
1
M
2
R
1
HMR1
(1,1)
80
80
320
(2,2)
258
277
1124
(3,3)
528
580
2360
(4,4)
890
989
4028
(5,5)
1344
1504
6128
(6,6)
1890
2125
8660
(7,7)
2528
2852
11 624
(8,8)
3258
3685
15 020
(9,9)
4080
4624
18 848
(10,10)
4994
5669
23 108Graphical explanation of Table1 is given in Figure 2.Figure 2
Comparison betweenM1, M2, and HM of N 6,6.


#### 2.1.4. ABC Index

(12)
ABC
R
1
=
∑
β
γ
∈
E
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
+
∑
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
+
∑
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
E
22
R
1
2
2
+
E
23
R
1
2
2
+
E
33
R
1
2
3
,
=
9
2
2
p
q
+
4
2
−
2
3
p
+
2
2
−
2
3
q
+
2
3
−
2
2
.


#### 2.1.5. GA Index

(13)
GA
R
1
=
∑
β
γ
∈
E
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
+
∑
β
γ
∈
E
23
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
+
∑
β
γ
∈
E
33
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
E
22
R
1
1
+
E
23
R
1
2
6
5
+
E
33
R
1
1
,
=
8
6
5
+
6
p
q
+
4
6
5
+
5
p
+
4
6
5
+
1
q
+
8
−
16
6
5
.Graphical explanation of Table3 is given in Figure 3.Table 3
Comparison of ABC and GA for N 6,6.
(p,q)
ABCR1
GAR1
(1,1)
10
2
20
(2,2)
59
2
/
2 + 2/3
32
6
/
5 + 44
(3,3)
58
2 + 8/3
16
6 + 80
(4,4)
191
2
/
2 + 6
144
6
/
5 + 128
(5,5)
142
2 + 32/3
224
6
/
5 + 188
(6,6)
395
2
/
2 + 50/3
64
6 + 260
(7,7)
262
2 + 24
432
6
/
5 + 344
(8,8)
671
2
/
2 + 98/3
112
6 + 440
(9,9)
418
2 + 128/3
704
6
/
5 + 548
(10,10)
1019
2
/
2 + 54
864
6
/
5 + 668Figure 3
Comparison between ABC and GA of N 6,6.


#### 2.1.6. Multiplicative First Zagreb Index

(14)
PM
1
R
1
=
∏
β
γ
∈
E
R
1
∂
β
+
∂
γ
,
=
∏
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
×
∏
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
×
∏
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
,
=
4
E
22
R
1
×
5
E
23
R
1
×
6
E
33
R
1
,
=
4
5
p
q
+
6
p
+
2
q
+
7
×
5
4
p
q
+
2
p
+
2
q
−
8
×
6
p
q
−
p
−
q
+
1
.


#### 2.1.7. Multiplicative Second Zagreb Index

(15)
PM
2
R
1
=
∏
β
γ
∈
E
R
1
∂
β
×
∂
γ
,
=
∏
β
γ
∈
E
22
R
1
∂
β
×
∂
γ
×
∏
β
γ
∈
E
23
R
1
∂
β
×
∂
γ
×
∏
β
γ
∈
E
33
R
1
∂
β
×
∂
γ
,
=
4
E
22
R
1
×
6
E
23
R
1
×
9
E
33
R
1
,
=
4
5
p
q
+
6
p
+
2
q
+
7
×
6
4
p
q
+
2
p
+
2
q
−
8
×
9
p
q
−
p
−
q
+
1
.


#### 2.1.8. First Zagreb Polynomial

(16)
M
1
R
1
,
x
=
∑
β
γ
∈
E
R
1
x
∂
β
+
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
x
∂
β
+
∂
γ
+
∑
β
γ
∈
E
23
R
1
x
∂
β
+
∂
γ
+
∑
β
γ
∈
E
33
R
1
x
∂
β
+
∂
γ
,
=
E
22
R
1
x
4
+
E
23
R
1
x
5
+
E
33
R
1
x
6
,
=
5
p
q
+
6
p
+
2
q
+
7
x
4
+
4
p
q
+
2
p
+
2
q
−
8
x
5
+
p
q
−
p
−
q
+
1
x
6
.


#### 2.1.9. Second Zagreb Polynomial

(17)
M
2
R
1
,
x
=
∑
β
γ
∈
E
R
1
x
∂
β
×
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
x
∂
β
×
∂
γ
+
∑
β
γ
∈
E
23
R
1
x
∂
β
×
∂
γ
+
∑
β
γ
∈
E
33
R
1
x
∂
β
×
∂
γ
,
=
E
22
R
1
x
4
+
E
23
R
1
x
6
+
E
33
R
1
x
9
,
=
5
p
q
+
6
p
+
2
q
+
7
x
4
+
4
p
q
+
2
p
+
2
q
−
8
x
6
+
p
q
−
p
−
q
+
1
x
9
.In Table4, the relation of the 1st Zagreb polynomial and 2nd Zagreb polynomial with each other is described.Table 4
Comparison ofM1R1,x and M2R1,x for N 6,6.
(p,q,x)
M
1
R
1
,
x
M
2
R
1
,
x
(1,1,1)
20
20
(2,2,1)
60
60
(3,3,1)
120
120
(4,4,1)
200
200
(5,5,1)
300
300
(6,6,1)
420
420
(7,7,1)
560
560
(8,8,1)
720
720
(9,9,1)
900
900
(10,10,1)
1100
1100Graphical explanation of Table4 is given in Figure 4.Figure 4
Comparison betweenM1R1,x and M2R1,x of N 6,6.


### 2.2. Topological Indices and Polynomials of Polythiophene

Theorem 2.
LetR2 be the graph of polythiophene, as shown in Figure 5. Then,Figure 5
Polythiophene.(18)
M
1
R
2
=
30
p
−
5
,
M
2
R
2
=
37
p
−
17
,
HM
R
2
=
152
p
−
72
,
PM
1
R
2
=
4
p
+
4
×
5
4
p
−
4
×
6
p
−
1
,
PM
2
R
2
=
4
p
+
4
×
6
4
p
−
4
×
9
p
−
1
,
M
1
R
2
,
x
=
p
−
1
x
6
+
4
p
−
4
x
5
+
p
+
4
x
4
,
M
2
R
2
,
x
=
p
−
1
x
9
+
4
p
−
4
x
6
+
p
+
4
x
4
.Proof.
LetR2 be isomorphic to the chemical graph polythiophene. Edge partition is described in Table 5 for the chemical graph polythiophene having p cycles with one connecting edge based on the degree sum of end vertices of each edge. The graph has total edges 6p − 1, and edge partition is given by(19)E2,2R2=β,γ∈ER2|∂β=2,∂γ=2,E2,3R2=β,γ∈ER2|∂β=2,∂γ=3,E3,3R2=β,γ∈ER2|∂β=3,∂γ=3.Table 5
Partition of the edges of polythiophene with respect to degrees of end nodes.
∂
β
,
∂
γ
E
∂
β
,
∂
γ
(2,2)
p + 4
(2,3)
4p − 4
(3,3)
p − 1


#### 2.2.1. First Zagreb Index

(20)
M
1
R
2
=
∑
β
γ
∈
E
R
2
∂
β
+
∂
γ
,
=
4
E
22
R
2
+
5
E
23
R
2
+
6
E
33
R
2
,
=
30
p
−
5.


#### 2.2.2. Second Zagreb Index

(21)
M
2
R
2
=
∑
β
γ
∈
E
R
2
∂
β
×
∂
γ
,
=
4
E
22
R
2
+
6
E
23
R
2
+
9
E
33
R
2
,
=
37
p
−
17.


#### 2.2.3. Hyper Zagreb Index

(22)
HM
R
2
=
∑
β
γ
∈
E
R
2
∂
β
+
∂
γ
2
,
=
4
2
E
22
R
2
+
5
2
E
23
R
2
+
6
2
E
33
R
2
,
=
152
p
−
72.In Table6, the relation of the 1st Zagreb index, 2nd Zagreb index, and hyper Zagreb index with one another for polythiophene is described.Table 6
Comparison ofM1, M2, and HM for polythiophene.
p
M
1
R
2
M
2
R
2
HMR2
1
25
20
80
2
55
57
272
3
85
94
384
4
115
131
536
5
145
168
688
6
175
205
840
7
205
242
992
8
235
279
1144
9
265
316
1296
10
295
353
1448Graphical explanation ofM1, M2, and HM of Table 6 is given in Figure 6.Figure 6
Comparison ofM1, M2, and HM of polythiophene.


#### 2.2.4. ABC Index

(23)
ABC
R
2
=
∑
β
γ
∈
E
R
2
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
E
22
R
2
2
2
+
E
23
R
2
2
2
+
E
33
R
2
2
3
,
=
5
2
2
+
2
3
p
+
2
3
.


#### 2.2.5. GA Index

(24)
GA
R
2
=
∑
β
γ
∈
E
R
2
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
E
22
R
2
1
+
E
23
R
2
2
6
5
+
E
33
R
2
1
,
=
8
6
5
+
2
p
+
5
−
8
6
5
.In Table7, the relation of the ABC index and GA index with each other for polythiophene is described.Table 7
Comparison of ABC and GA for polythiophene.
p
ABCR2
GAR2
1
5
2
/
2 + 4/3
7
2
5
2 + 2
8
6
/
5 + 9
3
15
2
/
2 + 8/3 + 2/3
16
6
/
5 + 11
4
10
2 + 10/3
24
6
/
5 + 13
5
25
2
/
2 + 4
32
6
/
5 + 15
6
15
2 + 14/3
8
6 + 17
7
35
2
/
2 + 16/3
48
6
/
5 + 19
8
20
2 + 6
56
6
/
5 + 21
9
45
2
/
2 + 20/3
64
6
/
5 + 23
10
25
2 + 22/3
72
6
/
5 + 25Graphical explanation of ABC and GA of polythiophene is given in Figure7.Figure 7
Comparison between ABC and GA of polythiophene.


#### 2.2.6. Multiplicative First Zagreb Index

(25)
PM
1
R
2
=
∏
β
γ
∈
E
R
2
∂
β
+
∂
γ
,
=
4
E
22
R
2
×
5
E
23
R
2
×
6
E
33
R
2
,
=
4
p
+
4
×
5
4
p
−
4
×
6
p
−
1
.


#### 2.2.7. Multiplicative Second Zagreb Index

(26)
PM
2
R
2
=
∏
β
γ
∈
E
R
2
∂
β
×
∂
γ
,
=
4
E
22
R
2
×
6
E
23
R
2
×
9
E
33
R
2
,
=
4
5
p
q
+
6
p
+
2
q
+
7
×
6
4
p
q
+
2
p
+
2
q
−
8
×
9
p
q
−
p
−
q
+
1
.


#### 2.2.8. First Zagreb Polynomial

(27)
M
1
R
2
,
x
=
∑
β
γ
∈
E
R
2
x
∂
β
+
∂
γ
,
=
E
22
R
2
x
4
+
E
23
R
2
x
5
+
E
33
R
2
x
6
,
=
p
+
4
x
4
+
4
p
−
4
x
5
+
p
−
1
x
6
.


#### 2.2.9. Second Zagreb Polynomial

(28)
M
2
R
2
,
x
=
∑
β
γ
∈
E
R
2
x
∂
β
×
∂
γ
,
=
E
22
R
2
x
4
+
E
23
R
2
x
6
+
E
33
R
2
x
9
,
=
p
+
4
x
4
+
4
p
−
4
x
6
+
p
−
1
x
9
.In Table8, the relation of the 1st Zagreb polynomial and 2nd Zagreb polynomial index with each other for polythiophene is described.Table 8
Comparison betweenM1R2,x and M2R2,x for polythiophene.
(p,x)
M
1
R
2
,
x
M
2
R
2
,
x
(1,1)
5
5
(2,1)
11
11
(3,1)
17
17
(4,1)
23
23
(5,1)
29
29
(6,1)
35
35
(7,1)
41
41
(8,1)
47
47
(9,1)
53
53
(10,1)
59
59Graphical explanation of Table8 is given in Figure 8.Figure 8
Comparison betweenM1R2,x and M2R2,x of polythiophene.


### 2.3. Topological Indices and Polynomials of the Backbone Structure of DNA

Theorem 3.
LetR3 be the graph of the backbone structure of DNA. Then,(29)M1R3=38p−18,M2R3=45p−26,HMR3=184p−106,PM1R3=4p+4×54p−4×6p−1,PM2R3=4p+4×64p−4×9p−1,M1R3,x=p−1x6+4p−4x5+p+4x4,M2R3,x=p−1x9+4p−4x6+p+4x4.Proof.
The graph is isomorphic to the backbone structure of DNA as shown in Figure9. There are p pentagon cycles connected with the p − 1 family of three edges in line, and edge partition is described in Table 9 for the graph based on the degree sum of end vertices of each edge. There are totally 8p − 3 edges, and edge partition is given by(30)E2,2R3=β,γ∈ER3|∂β=2,∂γ=2,E2,3R3=β,γ∈ER3|∂β=2,∂γ=3,E3,3R3=β,γ∈ER3|∂β=3,∂γ=3.Figure 9
Backbone structure of DNA.Table 9
Partition of the edges of DNA with respect to the degrees of end nodes.
∂
β
,
∂
γ
E
∂
β
,
∂
γ
(2,2)
3p + 1
(2,3)
4p − 2
(3,3)
p − 2


#### 2.3.1. First Zagreb Index

(31)
M
1
R
3
=
∑
β
γ
∈
E
R
3
∂
β
+
∂
γ
,
=
4
E
22
R
3
+
5
E
23
R
3
+
6
E
33
R
3
,
=
38
p
−
18.


#### 2.3.2. Second Zagreb Index

(32)
M
2
R
3
=
∑
β
γ
∈
E
R
3
∂
β
×
∂
γ
,
=
4
E
22
R
3
+
6
E
23
R
3
+
9
E
33
R
3
,
=
45
p
−
26.


#### 2.3.3. Hyper Zagreb Index

(33)
HM
R
3
=
∑
β
γ
∈
E
R
3
∂
β
+
∂
γ
2
,
=
4
2
E
22
R
3
+
5
2
E
23
R
3
+
6
2
E
33
R
3
,
=
184
p
−
106.In Table10, the relation of the 1st Zagreb index, 2nd Zagreb index, and hyper Zagreb index with one another for the backbone structure of DNA is described.Table 10
Comparison ofM1, M2, and HM for DNA.
p
M
1
R
3
M
2
R
3
HMR3
1
20
19
30
2
58
64
178
3
96
109
326
4
134
154
474
5
172
199
622
6
210
244
770
7
248
289
918
8
286
334
1066
9
324
379
1214
10
362
424
1362Graphical explanation of Table10 is given in Figure 10.Figure 10
Comparison ofM1R3, M2R3, and HM R3 of the given structure of DNA.


#### 2.3.4. ABC Index

(34)
ABC
R
3
=
∑
β
γ
∈
E
R
3
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
E
22
R
3
2
2
+
E
23
R
3
2
2
+
E
33
R
3
2
3
,
=
7
2
2
+
2
3
p
−
2
2
−
4
3
.


#### 2.3.5. GA Index

(35)
GA
R
3
=
∑
β
γ
∈
E
R
3
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
E
22
R
3
1
+
E
23
R
3
2
6
5
+
E
33
R
3
1
,
=
8
6
5
+
4
p
−
4
6
5
−
1.In Table11, the relation of the ABC index and GA index for the backbone structure of DNA is described.Table 11
Comparison of ABC and GA for DNA.
p
ABCR3
GAR3
1
3
2 − 2/3
4
6
/
5 + 3
2
13
2
/
2
12
6
/
5 + 7
3
10
2 + 2/3
4
6 + 11
4
27
2
/
2 + 4/3
28
6
/
5 + 15
5
17
2 + 2
36
6
/
5 + 19
6
41
2
/
2 + 8/3
44
6
/
5 + 23
7
24
2 + 10/3
52
6
/
5 + 27
8
55
2
/
2 + 4
12
6
/
5 + 31
9
31
2 + 14/3
68
6
/
5 + 35
10
69
2
/
2 + 16/3
76
6
/
5 + 39Graphical explanation of Table11 is given in Figure 11.Figure 11
Comparison between ABC and GA for DNA.


#### 2.3.6. Multiplicative First Zagreb Index

(36)
PM
1
R
3
=
∏
β
γ
∈
E
R
3
∂
β
+
∂
γ
,
=
4
E
22
R
3
×
5
E
23
R
3
×
6
E
33
R
3
,
=
4
3
p
+
1
×
5
4
p
−
2
×
6
p
−
2
.


#### 2.3.7. Multiplicative Second Zagreb Index

(37)
PM
2
R
2
=
∏
β
γ
∈
E
R
2
∂
β
×
∂
γ
,
=
4
E
22
R
3
×
6
E
23
R
3
×
9
E
33
R
3
,
=
4
3
p
+
1
×
6
4
p
−
2
×
9
p
−
2
.


#### 2.3.8. First Zagreb Polynomial

(38)
M
1
R
3
,
x
=
∑
β
γ
∈
E
R
3
x
∂
β
+
∂
γ
,
=
E
22
R
3
x
4
+
E
23
R
3
x
5
+
E
33
R
3
x
6
,
=
3
p
+
1
x
4
+
4
p
−
2
x
5
+
p
−
2
x
6
.


#### 2.3.9. Second Zagreb Polynomial

(39)
M
2
R
3
,
x
=
∑
β
γ
∈
E
R
3
x
∂
β
×
∂
γ
,
=
E
22
R
3
x
4
+
E
23
R
3
x
6
+
E
33
R
3
x
9
,
=
3
p
+
1
x
4
+
4
p
−
2
x
6
+
p
−
2
x
9
.In Table12, the relation of the 1st Zagreb polynomial and 2nd Zagreb polynomial index with each other for DNA is described.Table 12
Comparison ofM1R3,x and M2R3,x for DNA.
(p,x)
M
1
R
3
,
x
M
2
R
3
,
x
(1,1)
3
3
(2,1)
13
13
(3,1)
21
21
(4,1)
29
29
(5,1)
37
37
(6,1)
45
45
(7,1)
53
53
(8,1)
61
61
(9,1)
69
69
(10,1)
77
77Comparison ofM1R3,x and M2R3,x of Table 12 is given in Figure 12.Figure 12
Comparison betweenM1R3,x and M2R3,x of the backbone structure of DNA.


## 2.1. Topological Indices and Polynomials of Nylon6,6

Theorem 1.
LetR1 be the graph nylon 6,6, as shown in Figure 1. Then,Figure 1
Nylon 6,6.(7)
M
1
R
1
=
46
p
q
+
28
p
+
12
q
−
6
,
M
2
R
1
=
53
p
q
+
27
p
+
11
q
−
11
,
HM
R
1
=
216
p
q
+
110
p
+
46
q
−
52
,
PM
1
R
1
=
4
5
p
q
+
6
p
+
2
q
+
7
×
5
4
p
q
+
2
p
+
2
q
−
8
×
6
p
q
−
p
−
q
+
1
,
PM
2
R
1
=
4
5
p
q
+
6
p
+
2
q
+
7
×
6
4
p
q
+
2
p
+
2
q
−
8
×
9
p
q
−
p
−
q
+
1
,
M
1
R
1
,
x
=
5
p
q
+
6
p
+
2
q
+
7
x
4
+
4
p
q
+
2
p
+
2
q
−
8
x
5
+
p
q
−
p
−
q
+
1
x
6
,
M
2
R
1
,
x
=
5
p
q
+
6
p
+
2
q
+
7
x
4
+
4
p
q
+
2
p
+
2
q
−
8
x
6
+
p
q
−
p
−
q
+
1
x
9
.Proof.
R
1 is isomorphic to nylon 6,6. Total edges of R1 are 10pq + 7p + 3q while there are horizontally 10p + 3 edges and vertically up to q with 7p edges at the end in addition. Edge partition is described in Table 1 for the chemical graph nylon 6,6 based on the degree sum of end vertices of each edge.(8)E2,2R1=β,γ∈ER1|∂β=2,∂γ=2,E2,3R1=β,γ∈ER1|∂β=2,∂γ=3,E3,3R1=β,γ∈ER1|∂β=3,∂γ=3.Table 1
Partition of the edges of N 6,6(p,q) with respect to the degrees of end nodes.
∂
β
,
∂
γ
E
∂
β
,
∂
γ
(2,2)
5pq + 6p + 2q + 7
(2,3)
4pq+2p+2q − 8
(3,3)
pq − p − q + 1


### 2.1.1. First Zagreb Index

(9)
M
1
R
1
=
∑
β
γ
∈
E
R
1
∂
β
+
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
+
∑
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
+
∑
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
,
=
4
E
22
R
1
+
5
E
23
R
1
+
6
E
33
R
1
,
M
1
R
1
=
46
p
q
+
28
p
+
12
q
−
6.


### 2.1.2. Second Zagreb Index

(10)
M
2
R
1
=
∑
β
γ
∈
E
R
1
∂
β
×
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
∂
β
×
∂
γ
+
∑
β
γ
∈
E
23
R
1
∂
β
×
∂
γ
+
∑
β
γ
∈
E
33
R
1
∂
β
×
∂
γ
,
=
4
E
22
R
1
+
6
E
23
R
1
+
9
E
33
R
1
,
M
2
R
1
=
53
p
q
+
27
p
+
11
q
−
11.


### 2.1.3. Hyper Zagreb Index

(11)
HM
R
1
=
∑
β
γ
∈
E
R
1
∂
β
+
∂
γ
2
,
=
∑
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
2
+
∑
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
2
+
∑
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
2
,
=
4
2
E
22
R
1
+
5
2
E
23
R
1
+
6
2
E
33
R
1
,
=
216
p
q
+
110
p
+
46
q
−
52.In Table2, the relation of the 1st Zagreb index, 2nd Zagreb index, and hyper Zagreb index with one another is described.Table 2
Comparison ofM1, M2, and HM for N 6,6.
(p,q)
M
1
R
1
M
2
R
1
HMR1
(1,1)
80
80
320
(2,2)
258
277
1124
(3,3)
528
580
2360
(4,4)
890
989
4028
(5,5)
1344
1504
6128
(6,6)
1890
2125
8660
(7,7)
2528
2852
11 624
(8,8)
3258
3685
15 020
(9,9)
4080
4624
18 848
(10,10)
4994
5669
23 108Graphical explanation of Table1 is given in Figure 2.Figure 2
Comparison betweenM1, M2, and HM of N 6,6.


### 2.1.4. ABC Index

(12)
ABC
R
1
=
∑
β
γ
∈
E
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
+
∑
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
+
∑
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
E
22
R
1
2
2
+
E
23
R
1
2
2
+
E
33
R
1
2
3
,
=
9
2
2
p
q
+
4
2
−
2
3
p
+
2
2
−
2
3
q
+
2
3
−
2
2
.


### 2.1.5. GA Index

(13)
GA
R
1
=
∑
β
γ
∈
E
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
+
∑
β
γ
∈
E
23
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
+
∑
β
γ
∈
E
33
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
E
22
R
1
1
+
E
23
R
1
2
6
5
+
E
33
R
1
1
,
=
8
6
5
+
6
p
q
+
4
6
5
+
5
p
+
4
6
5
+
1
q
+
8
−
16
6
5
.Graphical explanation of Table3 is given in Figure 3.Table 3
Comparison of ABC and GA for N 6,6.
(p,q)
ABCR1
GAR1
(1,1)
10
2
20
(2,2)
59
2
/
2 + 2/3
32
6
/
5 + 44
(3,3)
58
2 + 8/3
16
6 + 80
(4,4)
191
2
/
2 + 6
144
6
/
5 + 128
(5,5)
142
2 + 32/3
224
6
/
5 + 188
(6,6)
395
2
/
2 + 50/3
64
6 + 260
(7,7)
262
2 + 24
432
6
/
5 + 344
(8,8)
671
2
/
2 + 98/3
112
6 + 440
(9,9)
418
2 + 128/3
704
6
/
5 + 548
(10,10)
1019
2
/
2 + 54
864
6
/
5 + 668Figure 3
Comparison between ABC and GA of N 6,6.


### 2.1.6. Multiplicative First Zagreb Index

(14)
PM
1
R
1
=
∏
β
γ
∈
E
R
1
∂
β
+
∂
γ
,
=
∏
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
×
∏
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
×
∏
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
,
=
4
E
22
R
1
×
5
E
23
R
1
×
6
E
33
R
1
,
=
4
5
p
q
+
6
p
+
2
q
+
7
×
5
4
p
q
+
2
p
+
2
q
−
8
×
6
p
q
−
p
−
q
+
1
.


### 2.1.7. Multiplicative Second Zagreb Index

(15)
PM
2
R
1
=
∏
β
γ
∈
E
R
1
∂
β
×
∂
γ
,
=
∏
β
γ
∈
E
22
R
1
∂
β
×
∂
γ
×
∏
β
γ
∈
E
23
R
1
∂
β
×
∂
γ
×
∏
β
γ
∈
E
33
R
1
∂
β
×
∂
γ
,
=
4
E
22
R
1
×
6
E
23
R
1
×
9
E
33
R
1
,
=
4
5
p
q
+
6
p
+
2
q
+
7
×
6
4
p
q
+
2
p
+
2
q
−
8
×
9
p
q
−
p
−
q
+
1
.


### 2.1.8. First Zagreb Polynomial

(16)
M
1
R
1
,
x
=
∑
β
γ
∈
E
R
1
x
∂
β
+
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
x
∂
β
+
∂
γ
+
∑
β
γ
∈
E
23
R
1
x
∂
β
+
∂
γ
+
∑
β
γ
∈
E
33
R
1
x
∂
β
+
∂
γ
,
=
E
22
R
1
x
4
+
E
23
R
1
x
5
+
E
33
R
1
x
6
,
=
5
p
q
+
6
p
+
2
q
+
7
x
4
+
4
p
q
+
2
p
+
2
q
−
8
x
5
+
p
q
−
p
−
q
+
1
x
6
.


### 2.1.9. Second Zagreb Polynomial

(17)
M
2
R
1
,
x
=
∑
β
γ
∈
E
R
1
x
∂
β
×
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
x
∂
β
×
∂
γ
+
∑
β
γ
∈
E
23
R
1
x
∂
β
×
∂
γ
+
∑
β
γ
∈
E
33
R
1
x
∂
β
×
∂
γ
,
=
E
22
R
1
x
4
+
E
23
R
1
x
6
+
E
33
R
1
x
9
,
=
5
p
q
+
6
p
+
2
q
+
7
x
4
+
4
p
q
+
2
p
+
2
q
−
8
x
6
+
p
q
−
p
−
q
+
1
x
9
.In Table4, the relation of the 1st Zagreb polynomial and 2nd Zagreb polynomial with each other is described.Table 4
Comparison ofM1R1,x and M2R1,x for N 6,6.
(p,q,x)
M
1
R
1
,
x
M
2
R
1
,
x
(1,1,1)
20
20
(2,2,1)
60
60
(3,3,1)
120
120
(4,4,1)
200
200
(5,5,1)
300
300
(6,6,1)
420
420
(7,7,1)
560
560
(8,8,1)
720
720
(9,9,1)
900
900
(10,10,1)
1100
1100Graphical explanation of Table4 is given in Figure 4.Figure 4
Comparison betweenM1R1,x and M2R1,x of N 6,6.


## 2.1.1. First Zagreb Index

(9)
M
1
R
1
=
∑
β
γ
∈
E
R
1
∂
β
+
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
+
∑
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
+
∑
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
,
=
4
E
22
R
1
+
5
E
23
R
1
+
6
E
33
R
1
,
M
1
R
1
=
46
p
q
+
28
p
+
12
q
−
6.


## 2.1.2. Second Zagreb Index

(10)
M
2
R
1
=
∑
β
γ
∈
E
R
1
∂
β
×
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
∂
β
×
∂
γ
+
∑
β
γ
∈
E
23
R
1
∂
β
×
∂
γ
+
∑
β
γ
∈
E
33
R
1
∂
β
×
∂
γ
,
=
4
E
22
R
1
+
6
E
23
R
1
+
9
E
33
R
1
,
M
2
R
1
=
53
p
q
+
27
p
+
11
q
−
11.


## 2.1.3. Hyper Zagreb Index

(11)
HM
R
1
=
∑
β
γ
∈
E
R
1
∂
β
+
∂
γ
2
,
=
∑
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
2
+
∑
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
2
+
∑
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
2
,
=
4
2
E
22
R
1
+
5
2
E
23
R
1
+
6
2
E
33
R
1
,
=
216
p
q
+
110
p
+
46
q
−
52.In Table2, the relation of the 1st Zagreb index, 2nd Zagreb index, and hyper Zagreb index with one another is described.Table 2
Comparison ofM1, M2, and HM for N 6,6.
(p,q)
M
1
R
1
M
2
R
1
HMR1
(1,1)
80
80
320
(2,2)
258
277
1124
(3,3)
528
580
2360
(4,4)
890
989
4028
(5,5)
1344
1504
6128
(6,6)
1890
2125
8660
(7,7)
2528
2852
11 624
(8,8)
3258
3685
15 020
(9,9)
4080
4624
18 848
(10,10)
4994
5669
23 108Graphical explanation of Table1 is given in Figure 2.Figure 2
Comparison betweenM1, M2, and HM of N 6,6.


## 2.1.4. ABC Index

(12)
ABC
R
1
=
∑
β
γ
∈
E
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
+
∑
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
+
∑
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
E
22
R
1
2
2
+
E
23
R
1
2
2
+
E
33
R
1
2
3
,
=
9
2
2
p
q
+
4
2
−
2
3
p
+
2
2
−
2
3
q
+
2
3
−
2
2
.


## 2.1.5. GA Index

(13)
GA
R
1
=
∑
β
γ
∈
E
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
+
∑
β
γ
∈
E
23
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
+
∑
β
γ
∈
E
33
R
1
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
E
22
R
1
1
+
E
23
R
1
2
6
5
+
E
33
R
1
1
,
=
8
6
5
+
6
p
q
+
4
6
5
+
5
p
+
4
6
5
+
1
q
+
8
−
16
6
5
.Graphical explanation of Table3 is given in Figure 3.Table 3
Comparison of ABC and GA for N 6,6.
(p,q)
ABCR1
GAR1
(1,1)
10
2
20
(2,2)
59
2
/
2 + 2/3
32
6
/
5 + 44
(3,3)
58
2 + 8/3
16
6 + 80
(4,4)
191
2
/
2 + 6
144
6
/
5 + 128
(5,5)
142
2 + 32/3
224
6
/
5 + 188
(6,6)
395
2
/
2 + 50/3
64
6 + 260
(7,7)
262
2 + 24
432
6
/
5 + 344
(8,8)
671
2
/
2 + 98/3
112
6 + 440
(9,9)
418
2 + 128/3
704
6
/
5 + 548
(10,10)
1019
2
/
2 + 54
864
6
/
5 + 668Figure 3
Comparison between ABC and GA of N 6,6.


## 2.1.6. Multiplicative First Zagreb Index

(14)
PM
1
R
1
=
∏
β
γ
∈
E
R
1
∂
β
+
∂
γ
,
=
∏
β
γ
∈
E
22
R
1
∂
β
+
∂
γ
×
∏
β
γ
∈
E
23
R
1
∂
β
+
∂
γ
×
∏
β
γ
∈
E
33
R
1
∂
β
+
∂
γ
,
=
4
E
22
R
1
×
5
E
23
R
1
×
6
E
33
R
1
,
=
4
5
p
q
+
6
p
+
2
q
+
7
×
5
4
p
q
+
2
p
+
2
q
−
8
×
6
p
q
−
p
−
q
+
1
.


## 2.1.7. Multiplicative Second Zagreb Index

(15)
PM
2
R
1
=
∏
β
γ
∈
E
R
1
∂
β
×
∂
γ
,
=
∏
β
γ
∈
E
22
R
1
∂
β
×
∂
γ
×
∏
β
γ
∈
E
23
R
1
∂
β
×
∂
γ
×
∏
β
γ
∈
E
33
R
1
∂
β
×
∂
γ
,
=
4
E
22
R
1
×
6
E
23
R
1
×
9
E
33
R
1
,
=
4
5
p
q
+
6
p
+
2
q
+
7
×
6
4
p
q
+
2
p
+
2
q
−
8
×
9
p
q
−
p
−
q
+
1
.


## 2.1.8. First Zagreb Polynomial

(16)
M
1
R
1
,
x
=
∑
β
γ
∈
E
R
1
x
∂
β
+
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
x
∂
β
+
∂
γ
+
∑
β
γ
∈
E
23
R
1
x
∂
β
+
∂
γ
+
∑
β
γ
∈
E
33
R
1
x
∂
β
+
∂
γ
,
=
E
22
R
1
x
4
+
E
23
R
1
x
5
+
E
33
R
1
x
6
,
=
5
p
q
+
6
p
+
2
q
+
7
x
4
+
4
p
q
+
2
p
+
2
q
−
8
x
5
+
p
q
−
p
−
q
+
1
x
6
.


## 2.1.9. Second Zagreb Polynomial

(17)
M
2
R
1
,
x
=
∑
β
γ
∈
E
R
1
x
∂
β
×
∂
γ
,
=
∑
β
γ
∈
E
22
R
1
x
∂
β
×
∂
γ
+
∑
β
γ
∈
E
23
R
1
x
∂
β
×
∂
γ
+
∑
β
γ
∈
E
33
R
1
x
∂
β
×
∂
γ
,
=
E
22
R
1
x
4
+
E
23
R
1
x
6
+
E
33
R
1
x
9
,
=
5
p
q
+
6
p
+
2
q
+
7
x
4
+
4
p
q
+
2
p
+
2
q
−
8
x
6
+
p
q
−
p
−
q
+
1
x
9
.In Table4, the relation of the 1st Zagreb polynomial and 2nd Zagreb polynomial with each other is described.Table 4
Comparison ofM1R1,x and M2R1,x for N 6,6.
(p,q,x)
M
1
R
1
,
x
M
2
R
1
,
x
(1,1,1)
20
20
(2,2,1)
60
60
(3,3,1)
120
120
(4,4,1)
200
200
(5,5,1)
300
300
(6,6,1)
420
420
(7,7,1)
560
560
(8,8,1)
720
720
(9,9,1)
900
900
(10,10,1)
1100
1100Graphical explanation of Table4 is given in Figure 4.Figure 4
Comparison betweenM1R1,x and M2R1,x of N 6,6.


## 2.2. Topological Indices and Polynomials of Polythiophene

Theorem 2.
LetR2 be the graph of polythiophene, as shown in Figure 5. Then,Figure 5
Polythiophene.(18)
M
1
R
2
=
30
p
−
5
,
M
2
R
2
=
37
p
−
17
,
HM
R
2
=
152
p
−
72
,
PM
1
R
2
=
4
p
+
4
×
5
4
p
−
4
×
6
p
−
1
,
PM
2
R
2
=
4
p
+
4
×
6
4
p
−
4
×
9
p
−
1
,
M
1
R
2
,
x
=
p
−
1
x
6
+
4
p
−
4
x
5
+
p
+
4
x
4
,
M
2
R
2
,
x
=
p
−
1
x
9
+
4
p
−
4
x
6
+
p
+
4
x
4
.Proof.
LetR2 be isomorphic to the chemical graph polythiophene. Edge partition is described in Table 5 for the chemical graph polythiophene having p cycles with one connecting edge based on the degree sum of end vertices of each edge. The graph has total edges 6p − 1, and edge partition is given by(19)E2,2R2=β,γ∈ER2|∂β=2,∂γ=2,E2,3R2=β,γ∈ER2|∂β=2,∂γ=3,E3,3R2=β,γ∈ER2|∂β=3,∂γ=3.Table 5
Partition of the edges of polythiophene with respect to degrees of end nodes.
∂
β
,
∂
γ
E
∂
β
,
∂
γ
(2,2)
p + 4
(2,3)
4p − 4
(3,3)
p − 1


### 2.2.1. First Zagreb Index

(20)
M
1
R
2
=
∑
β
γ
∈
E
R
2
∂
β
+
∂
γ
,
=
4
E
22
R
2
+
5
E
23
R
2
+
6
E
33
R
2
,
=
30
p
−
5.


### 2.2.2. Second Zagreb Index

(21)
M
2
R
2
=
∑
β
γ
∈
E
R
2
∂
β
×
∂
γ
,
=
4
E
22
R
2
+
6
E
23
R
2
+
9
E
33
R
2
,
=
37
p
−
17.


### 2.2.3. Hyper Zagreb Index

(22)
HM
R
2
=
∑
β
γ
∈
E
R
2
∂
β
+
∂
γ
2
,
=
4
2
E
22
R
2
+
5
2
E
23
R
2
+
6
2
E
33
R
2
,
=
152
p
−
72.In Table6, the relation of the 1st Zagreb index, 2nd Zagreb index, and hyper Zagreb index with one another for polythiophene is described.Table 6
Comparison ofM1, M2, and HM for polythiophene.
p
M
1
R
2
M
2
R
2
HMR2
1
25
20
80
2
55
57
272
3
85
94
384
4
115
131
536
5
145
168
688
6
175
205
840
7
205
242
992
8
235
279
1144
9
265
316
1296
10
295
353
1448Graphical explanation ofM1, M2, and HM of Table 6 is given in Figure 6.Figure 6
Comparison ofM1, M2, and HM of polythiophene.


### 2.2.4. ABC Index

(23)
ABC
R
2
=
∑
β
γ
∈
E
R
2
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
E
22
R
2
2
2
+
E
23
R
2
2
2
+
E
33
R
2
2
3
,
=
5
2
2
+
2
3
p
+
2
3
.


### 2.2.5. GA Index

(24)
GA
R
2
=
∑
β
γ
∈
E
R
2
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
E
22
R
2
1
+
E
23
R
2
2
6
5
+
E
33
R
2
1
,
=
8
6
5
+
2
p
+
5
−
8
6
5
.In Table7, the relation of the ABC index and GA index with each other for polythiophene is described.Table 7
Comparison of ABC and GA for polythiophene.
p
ABCR2
GAR2
1
5
2
/
2 + 4/3
7
2
5
2 + 2
8
6
/
5 + 9
3
15
2
/
2 + 8/3 + 2/3
16
6
/
5 + 11
4
10
2 + 10/3
24
6
/
5 + 13
5
25
2
/
2 + 4
32
6
/
5 + 15
6
15
2 + 14/3
8
6 + 17
7
35
2
/
2 + 16/3
48
6
/
5 + 19
8
20
2 + 6
56
6
/
5 + 21
9
45
2
/
2 + 20/3
64
6
/
5 + 23
10
25
2 + 22/3
72
6
/
5 + 25Graphical explanation of ABC and GA of polythiophene is given in Figure7.Figure 7
Comparison between ABC and GA of polythiophene.


### 2.2.6. Multiplicative First Zagreb Index

(25)
PM
1
R
2
=
∏
β
γ
∈
E
R
2
∂
β
+
∂
γ
,
=
4
E
22
R
2
×
5
E
23
R
2
×
6
E
33
R
2
,
=
4
p
+
4
×
5
4
p
−
4
×
6
p
−
1
.


### 2.2.7. Multiplicative Second Zagreb Index

(26)
PM
2
R
2
=
∏
β
γ
∈
E
R
2
∂
β
×
∂
γ
,
=
4
E
22
R
2
×
6
E
23
R
2
×
9
E
33
R
2
,
=
4
5
p
q
+
6
p
+
2
q
+
7
×
6
4
p
q
+
2
p
+
2
q
−
8
×
9
p
q
−
p
−
q
+
1
.


### 2.2.8. First Zagreb Polynomial

(27)
M
1
R
2
,
x
=
∑
β
γ
∈
E
R
2
x
∂
β
+
∂
γ
,
=
E
22
R
2
x
4
+
E
23
R
2
x
5
+
E
33
R
2
x
6
,
=
p
+
4
x
4
+
4
p
−
4
x
5
+
p
−
1
x
6
.


### 2.2.9. Second Zagreb Polynomial

(28)
M
2
R
2
,
x
=
∑
β
γ
∈
E
R
2
x
∂
β
×
∂
γ
,
=
E
22
R
2
x
4
+
E
23
R
2
x
6
+
E
33
R
2
x
9
,
=
p
+
4
x
4
+
4
p
−
4
x
6
+
p
−
1
x
9
.In Table8, the relation of the 1st Zagreb polynomial and 2nd Zagreb polynomial index with each other for polythiophene is described.Table 8
Comparison betweenM1R2,x and M2R2,x for polythiophene.
(p,x)
M
1
R
2
,
x
M
2
R
2
,
x
(1,1)
5
5
(2,1)
11
11
(3,1)
17
17
(4,1)
23
23
(5,1)
29
29
(6,1)
35
35
(7,1)
41
41
(8,1)
47
47
(9,1)
53
53
(10,1)
59
59Graphical explanation of Table8 is given in Figure 8.Figure 8
Comparison betweenM1R2,x and M2R2,x of polythiophene.


## 2.2.1. First Zagreb Index

(20)
M
1
R
2
=
∑
β
γ
∈
E
R
2
∂
β
+
∂
γ
,
=
4
E
22
R
2
+
5
E
23
R
2
+
6
E
33
R
2
,
=
30
p
−
5.


## 2.2.2. Second Zagreb Index

(21)
M
2
R
2
=
∑
β
γ
∈
E
R
2
∂
β
×
∂
γ
,
=
4
E
22
R
2
+
6
E
23
R
2
+
9
E
33
R
2
,
=
37
p
−
17.


## 2.2.3. Hyper Zagreb Index

(22)
HM
R
2
=
∑
β
γ
∈
E
R
2
∂
β
+
∂
γ
2
,
=
4
2
E
22
R
2
+
5
2
E
23
R
2
+
6
2
E
33
R
2
,
=
152
p
−
72.In Table6, the relation of the 1st Zagreb index, 2nd Zagreb index, and hyper Zagreb index with one another for polythiophene is described.Table 6
Comparison ofM1, M2, and HM for polythiophene.
p
M
1
R
2
M
2
R
2
HMR2
1
25
20
80
2
55
57
272
3
85
94
384
4
115
131
536
5
145
168
688
6
175
205
840
7
205
242
992
8
235
279
1144
9
265
316
1296
10
295
353
1448Graphical explanation ofM1, M2, and HM of Table 6 is given in Figure 6.Figure 6
Comparison ofM1, M2, and HM of polythiophene.


## 2.2.4. ABC Index

(23)
ABC
R
2
=
∑
β
γ
∈
E
R
2
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
E
22
R
2
2
2
+
E
23
R
2
2
2
+
E
33
R
2
2
3
,
=
5
2
2
+
2
3
p
+
2
3
.


## 2.2.5. GA Index

(24)
GA
R
2
=
∑
β
γ
∈
E
R
2
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
E
22
R
2
1
+
E
23
R
2
2
6
5
+
E
33
R
2
1
,
=
8
6
5
+
2
p
+
5
−
8
6
5
.In Table7, the relation of the ABC index and GA index with each other for polythiophene is described.Table 7
Comparison of ABC and GA for polythiophene.
p
ABCR2
GAR2
1
5
2
/
2 + 4/3
7
2
5
2 + 2
8
6
/
5 + 9
3
15
2
/
2 + 8/3 + 2/3
16
6
/
5 + 11
4
10
2 + 10/3
24
6
/
5 + 13
5
25
2
/
2 + 4
32
6
/
5 + 15
6
15
2 + 14/3
8
6 + 17
7
35
2
/
2 + 16/3
48
6
/
5 + 19
8
20
2 + 6
56
6
/
5 + 21
9
45
2
/
2 + 20/3
64
6
/
5 + 23
10
25
2 + 22/3
72
6
/
5 + 25Graphical explanation of ABC and GA of polythiophene is given in Figure7.Figure 7
Comparison between ABC and GA of polythiophene.


## 2.2.6. Multiplicative First Zagreb Index

(25)
PM
1
R
2
=
∏
β
γ
∈
E
R
2
∂
β
+
∂
γ
,
=
4
E
22
R
2
×
5
E
23
R
2
×
6
E
33
R
2
,
=
4
p
+
4
×
5
4
p
−
4
×
6
p
−
1
.


## 2.2.7. Multiplicative Second Zagreb Index

(26)
PM
2
R
2
=
∏
β
γ
∈
E
R
2
∂
β
×
∂
γ
,
=
4
E
22
R
2
×
6
E
23
R
2
×
9
E
33
R
2
,
=
4
5
p
q
+
6
p
+
2
q
+
7
×
6
4
p
q
+
2
p
+
2
q
−
8
×
9
p
q
−
p
−
q
+
1
.


## 2.2.8. First Zagreb Polynomial

(27)
M
1
R
2
,
x
=
∑
β
γ
∈
E
R
2
x
∂
β
+
∂
γ
,
=
E
22
R
2
x
4
+
E
23
R
2
x
5
+
E
33
R
2
x
6
,
=
p
+
4
x
4
+
4
p
−
4
x
5
+
p
−
1
x
6
.


## 2.2.9. Second Zagreb Polynomial

(28)
M
2
R
2
,
x
=
∑
β
γ
∈
E
R
2
x
∂
β
×
∂
γ
,
=
E
22
R
2
x
4
+
E
23
R
2
x
6
+
E
33
R
2
x
9
,
=
p
+
4
x
4
+
4
p
−
4
x
6
+
p
−
1
x
9
.In Table8, the relation of the 1st Zagreb polynomial and 2nd Zagreb polynomial index with each other for polythiophene is described.Table 8
Comparison betweenM1R2,x and M2R2,x for polythiophene.
(p,x)
M
1
R
2
,
x
M
2
R
2
,
x
(1,1)
5
5
(2,1)
11
11
(3,1)
17
17
(4,1)
23
23
(5,1)
29
29
(6,1)
35
35
(7,1)
41
41
(8,1)
47
47
(9,1)
53
53
(10,1)
59
59Graphical explanation of Table8 is given in Figure 8.Figure 8
Comparison betweenM1R2,x and M2R2,x of polythiophene.


## 2.3. Topological Indices and Polynomials of the Backbone Structure of DNA

Theorem 3.
LetR3 be the graph of the backbone structure of DNA. Then,(29)M1R3=38p−18,M2R3=45p−26,HMR3=184p−106,PM1R3=4p+4×54p−4×6p−1,PM2R3=4p+4×64p−4×9p−1,M1R3,x=p−1x6+4p−4x5+p+4x4,M2R3,x=p−1x9+4p−4x6+p+4x4.Proof.
The graph is isomorphic to the backbone structure of DNA as shown in Figure9. There are p pentagon cycles connected with the p − 1 family of three edges in line, and edge partition is described in Table 9 for the graph based on the degree sum of end vertices of each edge. There are totally 8p − 3 edges, and edge partition is given by(30)E2,2R3=β,γ∈ER3|∂β=2,∂γ=2,E2,3R3=β,γ∈ER3|∂β=2,∂γ=3,E3,3R3=β,γ∈ER3|∂β=3,∂γ=3.Figure 9
Backbone structure of DNA.Table 9
Partition of the edges of DNA with respect to the degrees of end nodes.
∂
β
,
∂
γ
E
∂
β
,
∂
γ
(2,2)
3p + 1
(2,3)
4p − 2
(3,3)
p − 2


### 2.3.1. First Zagreb Index

(31)
M
1
R
3
=
∑
β
γ
∈
E
R
3
∂
β
+
∂
γ
,
=
4
E
22
R
3
+
5
E
23
R
3
+
6
E
33
R
3
,
=
38
p
−
18.


### 2.3.2. Second Zagreb Index

(32)
M
2
R
3
=
∑
β
γ
∈
E
R
3
∂
β
×
∂
γ
,
=
4
E
22
R
3
+
6
E
23
R
3
+
9
E
33
R
3
,
=
45
p
−
26.


### 2.3.3. Hyper Zagreb Index

(33)
HM
R
3
=
∑
β
γ
∈
E
R
3
∂
β
+
∂
γ
2
,
=
4
2
E
22
R
3
+
5
2
E
23
R
3
+
6
2
E
33
R
3
,
=
184
p
−
106.In Table10, the relation of the 1st Zagreb index, 2nd Zagreb index, and hyper Zagreb index with one another for the backbone structure of DNA is described.Table 10
Comparison ofM1, M2, and HM for DNA.
p
M
1
R
3
M
2
R
3
HMR3
1
20
19
30
2
58
64
178
3
96
109
326
4
134
154
474
5
172
199
622
6
210
244
770
7
248
289
918
8
286
334
1066
9
324
379
1214
10
362
424
1362Graphical explanation of Table10 is given in Figure 10.Figure 10
Comparison ofM1R3, M2R3, and HM R3 of the given structure of DNA.


### 2.3.4. ABC Index

(34)
ABC
R
3
=
∑
β
γ
∈
E
R
3
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
E
22
R
3
2
2
+
E
23
R
3
2
2
+
E
33
R
3
2
3
,
=
7
2
2
+
2
3
p
−
2
2
−
4
3
.


### 2.3.5. GA Index

(35)
GA
R
3
=
∑
β
γ
∈
E
R
3
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
E
22
R
3
1
+
E
23
R
3
2
6
5
+
E
33
R
3
1
,
=
8
6
5
+
4
p
−
4
6
5
−
1.In Table11, the relation of the ABC index and GA index for the backbone structure of DNA is described.Table 11
Comparison of ABC and GA for DNA.
p
ABCR3
GAR3
1
3
2 − 2/3
4
6
/
5 + 3
2
13
2
/
2
12
6
/
5 + 7
3
10
2 + 2/3
4
6 + 11
4
27
2
/
2 + 4/3
28
6
/
5 + 15
5
17
2 + 2
36
6
/
5 + 19
6
41
2
/
2 + 8/3
44
6
/
5 + 23
7
24
2 + 10/3
52
6
/
5 + 27
8
55
2
/
2 + 4
12
6
/
5 + 31
9
31
2 + 14/3
68
6
/
5 + 35
10
69
2
/
2 + 16/3
76
6
/
5 + 39Graphical explanation of Table11 is given in Figure 11.Figure 11
Comparison between ABC and GA for DNA.


### 2.3.6. Multiplicative First Zagreb Index

(36)
PM
1
R
3
=
∏
β
γ
∈
E
R
3
∂
β
+
∂
γ
,
=
4
E
22
R
3
×
5
E
23
R
3
×
6
E
33
R
3
,
=
4
3
p
+
1
×
5
4
p
−
2
×
6
p
−
2
.


### 2.3.7. Multiplicative Second Zagreb Index

(37)
PM
2
R
2
=
∏
β
γ
∈
E
R
2
∂
β
×
∂
γ
,
=
4
E
22
R
3
×
6
E
23
R
3
×
9
E
33
R
3
,
=
4
3
p
+
1
×
6
4
p
−
2
×
9
p
−
2
.


### 2.3.8. First Zagreb Polynomial

(38)
M
1
R
3
,
x
=
∑
β
γ
∈
E
R
3
x
∂
β
+
∂
γ
,
=
E
22
R
3
x
4
+
E
23
R
3
x
5
+
E
33
R
3
x
6
,
=
3
p
+
1
x
4
+
4
p
−
2
x
5
+
p
−
2
x
6
.


### 2.3.9. Second Zagreb Polynomial

(39)
M
2
R
3
,
x
=
∑
β
γ
∈
E
R
3
x
∂
β
×
∂
γ
,
=
E
22
R
3
x
4
+
E
23
R
3
x
6
+
E
33
R
3
x
9
,
=
3
p
+
1
x
4
+
4
p
−
2
x
6
+
p
−
2
x
9
.In Table12, the relation of the 1st Zagreb polynomial and 2nd Zagreb polynomial index with each other for DNA is described.Table 12
Comparison ofM1R3,x and M2R3,x for DNA.
(p,x)
M
1
R
3
,
x
M
2
R
3
,
x
(1,1)
3
3
(2,1)
13
13
(3,1)
21
21
(4,1)
29
29
(5,1)
37
37
(6,1)
45
45
(7,1)
53
53
(8,1)
61
61
(9,1)
69
69
(10,1)
77
77Comparison ofM1R3,x and M2R3,x of Table 12 is given in Figure 12.Figure 12
Comparison betweenM1R3,x and M2R3,x of the backbone structure of DNA.


## 2.3.1. First Zagreb Index

(31)
M
1
R
3
=
∑
β
γ
∈
E
R
3
∂
β
+
∂
γ
,
=
4
E
22
R
3
+
5
E
23
R
3
+
6
E
33
R
3
,
=
38
p
−
18.


## 2.3.2. Second Zagreb Index

(32)
M
2
R
3
=
∑
β
γ
∈
E
R
3
∂
β
×
∂
γ
,
=
4
E
22
R
3
+
6
E
23
R
3
+
9
E
33
R
3
,
=
45
p
−
26.


## 2.3.3. Hyper Zagreb Index

(33)
HM
R
3
=
∑
β
γ
∈
E
R
3
∂
β
+
∂
γ
2
,
=
4
2
E
22
R
3
+
5
2
E
23
R
3
+
6
2
E
33
R
3
,
=
184
p
−
106.In Table10, the relation of the 1st Zagreb index, 2nd Zagreb index, and hyper Zagreb index with one another for the backbone structure of DNA is described.Table 10
Comparison ofM1, M2, and HM for DNA.
p
M
1
R
3
M
2
R
3
HMR3
1
20
19
30
2
58
64
178
3
96
109
326
4
134
154
474
5
172
199
622
6
210
244
770
7
248
289
918
8
286
334
1066
9
324
379
1214
10
362
424
1362Graphical explanation of Table10 is given in Figure 10.Figure 10
Comparison ofM1R3, M2R3, and HM R3 of the given structure of DNA.


## 2.3.4. ABC Index

(34)
ABC
R
3
=
∑
β
γ
∈
E
R
3
∂
β
+
∂
γ
−
2
∂
β
∂
γ
,
=
E
22
R
3
2
2
+
E
23
R
3
2
2
+
E
33
R
3
2
3
,
=
7
2
2
+
2
3
p
−
2
2
−
4
3
.


## 2.3.5. GA Index

(35)
GA
R
3
=
∑
β
γ
∈
E
R
3
2
∂
β
×
∂
γ
∂
β
+
∂
γ
,
=
E
22
R
3
1
+
E
23
R
3
2
6
5
+
E
33
R
3
1
,
=
8
6
5
+
4
p
−
4
6
5
−
1.In Table11, the relation of the ABC index and GA index for the backbone structure of DNA is described.Table 11
Comparison of ABC and GA for DNA.
p
ABCR3
GAR3
1
3
2 − 2/3
4
6
/
5 + 3
2
13
2
/
2
12
6
/
5 + 7
3
10
2 + 2/3
4
6 + 11
4
27
2
/
2 + 4/3
28
6
/
5 + 15
5
17
2 + 2
36
6
/
5 + 19
6
41
2
/
2 + 8/3
44
6
/
5 + 23
7
24
2 + 10/3
52
6
/
5 + 27
8
55
2
/
2 + 4
12
6
/
5 + 31
9
31
2 + 14/3
68
6
/
5 + 35
10
69
2
/
2 + 16/3
76
6
/
5 + 39Graphical explanation of Table11 is given in Figure 11.Figure 11
Comparison between ABC and GA for DNA.


## 2.3.6. Multiplicative First Zagreb Index

(36)
PM
1
R
3
=
∏
β
γ
∈
E
R
3
∂
β
+
∂
γ
,
=
4
E
22
R
3
×
5
E
23
R
3
×
6
E
33
R
3
,
=
4
3
p
+
1
×
5
4
p
−
2
×
6
p
−
2
.


## 2.3.7. Multiplicative Second Zagreb Index

(37)
PM
2
R
2
=
∏
β
γ
∈
E
R
2
∂
β
×
∂
γ
,
=
4
E
22
R
3
×
6
E
23
R
3
×
9
E
33
R
3
,
=
4
3
p
+
1
×
6
4
p
−
2
×
9
p
−
2
.


## 2.3.8. First Zagreb Polynomial

(38)
M
1
R
3
,
x
=
∑
β
γ
∈
E
R
3
x
∂
β
+
∂
γ
,
=
E
22
R
3
x
4
+
E
23
R
3
x
5
+
E
33
R
3
x
6
,
=
3
p
+
1
x
4
+
4
p
−
2
x
5
+
p
−
2
x
6
.


## 2.3.9. Second Zagreb Polynomial

(39)
M
2
R
3
,
x
=
∑
β
γ
∈
E
R
3
x
∂
β
×
∂
γ
,
=
E
22
R
3
x
4
+
E
23
R
3
x
6
+
E
33
R
3
x
9
,
=
3
p
+
1
x
4
+
4
p
−
2
x
6
+
p
−
2
x
9
.In Table12, the relation of the 1st Zagreb polynomial and 2nd Zagreb polynomial index with each other for DNA is described.Table 12
Comparison ofM1R3,x and M2R3,x for DNA.
(p,x)
M
1
R
3
,
x
M
2
R
3
,
x
(1,1)
3
3
(2,1)
13
13
(3,1)
21
21
(4,1)
29
29
(5,1)
37
37
(6,1)
45
45
(7,1)
53
53
(8,1)
61
61
(9,1)
69
69
(10,1)
77
77Comparison ofM1R3,x and M2R3,x of Table 12 is given in Figure 12.Figure 12
Comparison betweenM1R3,x and M2R3,x of the backbone structure of DNA.


## 3. Conclusions

In this article, we computed the first Zagreb index, second Zagreb index, hyper Zagreb index, atom bond connectivity index, geometric-arithmetic index, first Zagreb polynomial, and second Zagreb polynomial of the grid graph nylon 6,6, chemical graph polythiophene, and backbone structure of DNA. We expanded nylon 6,6 up top and q cycles and polythiophene and backbone DNA structure up to p cycles. By using these results, the correlation of biological activities and physicochemical properties with chemical structure is very easy. Also, we gave a comparison for the different versions of the computed TIs by numerical values in the tables. Finding indices and polynomials of polythiophene and DNA structure up to p and q cycles is an open question for researchers.



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*Source: 8718807-2021-12-28.xml*