# Computation of New Degree-Based Topological Indices of Graphene

Authors: V. S. Shigehalli; Rachanna Kanabur
Journal: Journal of Mathematics
 (2016)

---

## Abstract

Graphene is one of the most promising nanomaterials because of its unique combination of superb properties, which opens a way for its exploitation in a wide spectrum of applications ranging from electronics to optics, sensors, and biodevices. Inspired by recent work on Graphene of computing topological indices, here we propose new topological indices, namely, Arithmetic-Geometric index (A
G
1 index), SK index, SK1 index, and SK2 index of a molecular graph G and obtain the explicit formulae of these indices for Graphene.


---

## Body

## 1. Introduction

A topological index of a chemical compound is an integer, derived following a certain rule, which can be used to characterize the chemical compound and predict certain physiochemical properties like boiling point, molecular weight, density, refractive index, and so forth [1, 2].A molecular graphG
=
(
V
,
E
) is a simple graph having n
=
V vertices and m
=
E edges. The vertices v
i
∈
V represent nonhydrogen atoms and the edges (
v
i
,
v
j
)
∈
E represent covalent bonds between the corresponding atoms. In particular, hydrocarbons are formed only by carbon and hydrogen atom and their molecular graphs represent the carbon skeleton of the molecule [1, 2].Molecular graphs are a special type of chemical graphs, which represent the constitution of molecules. They are also called constitutional graphs. When the constitutional graph of a molecule is represented in a two-dimensional basis, it is called structural graph [1, 2].All molecular graphs considered in this paper are finite, connected, loopless, and without multiple edges. LetG
=
(
V
,
E
) be a graph with n vertices and m edges. The degree of a vertex u
∈
V
(
G
) is denoted by d
u and is the number of vertices that are adjacent to u. The edge connecting the vertices u and v is denoted by u
v [3].


## 2. Computing the Topological Indices of Graphene

Graphene is an atomic scale honeycomb lattice made of carbon atoms. Graphene is 200 times stronger than steel, one million times thinner than a human hair, and world’s most conductive material. So it has captured the attention of scientists, researchers, and industrialists worldwide. It is one of the most promising nanomaterials because of its unique combination of superb properties, which opens a way for its exploitation in a wide spectrum of applications ranging from electronics to optics, sensors, and biodevices. Also it is the most effective material for electromagnetic interference (EMI) shielding. Now we focus on computation of topological indices of Graphene [4–6].Motivated by previous research on Graphene, here we introduce four new topological indices and computed their corresponding topological index value of Graphene [7–13].In Figure1, the molecular graph of Graphene is shown.Figure 1


### 2.1. Motivation

By looking at the earlier results for computing the topological indices for Graphene, here we introduce new degree-based topological indices to compute their values for Graphene.In the upcoming sections, topological indices and their computation of topological indices for Graphene are discussed.Definition 1 (Arithmetic-Geometric (AG1) index).
LetG
=
(
V
,
E
) be a molecular graph and d
u be the degree of the vertex u; then AG1 index of G is defined as(1)
A
G
1
G
=
∑
u
,
v
∈
E
G
d
G
u
+
d
G
v
2
d
G
u
·
d
G
v
,where A
G
1 index is considered for distinct vertices.
The above equation is the sum of the ratio of the Arithmetic mean and Geometric mean ofu and v, where d
G
(
u
) (or d
G
(
v
)
) denote the degree of the vertex u (or v).Definition 2 (SK index).
The SK index of a graphG
=
(
V
,
E
) is defined as S
K
(
G
)
=
∑
u
,
v
∈
E
(
G
)
(
(
d
G
u
+
d
G
v
)
/
2
), where d
G
(
u
) and d
G
(
v
) are the degrees of the vertices u and v in G, respectively.Definition 3 (SK1 index).
TheS
K
1 index of a graph G
=
(
V
,
E
) is defined as S
K
1
(
G
)
=
∑
u
,
v
∈
E
(
G
)
(
(
d
G
u
·
d
G
v
)
/
2
), where d
G
(
u
) and d
G
(
v
) are the product of the degrees of the vertices u and v in G, respectively.Definition 4 (SK2 index).
The SK2 index of a graph G
=
(
V
,
E
) is defined as S
K
2
(
G
)
=
∑
u
,
v
∈
E
(
G
)
d
G
u
+
d
G
v
/
2
2, where d
G
(
u
) and d
G
(
v
) are the degrees of the vertices u and v in G, respectively.


## 2.1. Motivation

By looking at the earlier results for computing the topological indices for Graphene, here we introduce new degree-based topological indices to compute their values for Graphene.In the upcoming sections, topological indices and their computation of topological indices for Graphene are discussed.Definition 1 (Arithmetic-Geometric (AG1) index).
LetG
=
(
V
,
E
) be a molecular graph and d
u be the degree of the vertex u; then AG1 index of G is defined as(1)
A
G
1
G
=
∑
u
,
v
∈
E
G
d
G
u
+
d
G
v
2
d
G
u
·
d
G
v
,where A
G
1 index is considered for distinct vertices.
The above equation is the sum of the ratio of the Arithmetic mean and Geometric mean ofu and v, where d
G
(
u
) (or d
G
(
v
)
) denote the degree of the vertex u (or v).Definition 2 (SK index).
The SK index of a graphG
=
(
V
,
E
) is defined as S
K
(
G
)
=
∑
u
,
v
∈
E
(
G
)
(
(
d
G
u
+
d
G
v
)
/
2
), where d
G
(
u
) and d
G
(
v
) are the degrees of the vertices u and v in G, respectively.Definition 3 (SK1 index).
TheS
K
1 index of a graph G
=
(
V
,
E
) is defined as S
K
1
(
G
)
=
∑
u
,
v
∈
E
(
G
)
(
(
d
G
u
·
d
G
v
)
/
2
), where d
G
(
u
) and d
G
(
v
) are the product of the degrees of the vertices u and v in G, respectively.Definition 4 (SK2 index).
The SK2 index of a graph G
=
(
V
,
E
) is defined as S
K
2
(
G
)
=
∑
u
,
v
∈
E
(
G
)
d
G
u
+
d
G
v
/
2
2, where d
G
(
u
) and d
G
(
v
) are the degrees of the vertices u and v in G, respectively.


## 3. Main Results

Theorem 5.
TheA
G
1 index of Graphene having “t” rows of Benzene rings with “s” Benzene rings in each row is given by(2)
A
G
1
G
=
6
6
s
t
+
20
-
4
6
s
+
10
t
-
20
-
6
6
2
6
,
i
f
t
≠
1
2
6
+
20
s
+
6
6
-
10
2
6
,
i
f
t
=
1
.Proof.
Consider a Graphene having “t” rows with “s” Benzene rings in each row. Let m
i
,
j denote the number of edges connecting the vertices of degrees d
i and d
j. Two-dimensional structure of Graphene (Figure 1) contains only m
2
,
2, m
2,3, and m
3,3 edges. The number of m
2,2, m
2,3, and m
3,3 edge in each row is mentioned in Table 1.
Therefore Graphene containsm
2,2
=
(
t
+
4
) edges, m
2,3
=
(
4
s
+
2
t
-
4
) edges, and m
3,3
=
(
3
t
s
-
2
s
-
t
-
1
) edges.(3)
A
G
1
G
=
∑
u
,
v
∈
E
G
d
G
u
+
d
G
v
2
d
G
u
·
d
G
v
,
A
G
1
G
=
m
2,2
2
+
2
2
2.2
+
m
2,3
2
+
3
2
2.3
+
m
3,3
3
+
3
2
3.3
=
t
+
4
4
4
+
4
s
+
2
t
-
4
5
2
6
+
3
t
s
-
2
s
-
t
-
1
6
6
=
t
+
4
+
4
s
+
2
t
-
4
5
2
6
+
3
t
s
-
2
s
-
t
-
1
=
3
t
s
-
2
s
+
3
+
4
s
+
2
t
-
4
5
2
6
=
2
6
3
t
s
-
2
s
+
3
+
5
4
s
+
2
t
-
4
2
6
=
6
6
t
s
-
4
6
s
+
6
6
+
20
s
+
10
t
-
20
2
6
=
6
6
+
20
-
4
6
s
+
10
t
-
20
-
6
6
2
6
.Now consider the following cases.
Case
1. The Arithmetic-Geometric index of Graphene for t
≠
1 is (4)
A
G
1
G
=
6
6
+
20
-
4
6
s
+
10
t
-
20
-
6
6
2
6
.
Case
2. t
=
1, m
2,2
=
t
+
4, m
2,3
=
4
s
-
2, and m
3,3
=
s
-
2, edges as shown in Figure 2:(5)
A
G
1
G
=
∑
u
,
v
∈
E
G
d
G
u
+
d
G
v
2
d
G
u
·
d
G
v
,
A
G
1
G
=
m
2,2
2
+
2
2
2.2
+
m
2,3
2
+
3
2
2.3
+
m
3,3
3
+
3
2
3.3
=
t
+
4
4
4
+
4
s
+
2
t
-
4
5
2
6
+
3
t
s
-
2
s
-
t
-
1
6
6
=
5
1
+
4
s
-
2
5
2
6
+
s
-
2
1
=
s
+
3
+
4
s
-
2
5
2
6
=
2
6
s
+
6
6
+
20
s
-
10
2
6
=
2
6
+
20
s
+
6
6
-
10
2
6
.Table 1
Row
m
2,2
m
2,3
m
3,3
1
3
2
s
3
s
-
2
2
1
2
3
s
-
1
3
1
2
3
s
-
1
4
1
2
3
s
-
1
⋮
⋮
⋮
⋮
t
3
3
s
s
-
1
Total
t
+
4
4
s
+
2
t
-
4
3
t
s
-
2
s
-
t
-
1Figure 2Theorem 6.
The SK index of Graphene having “t” rows of Benzene rings with “s” Benzene rings in each row is given by(6)
SK
G
=
18
t
s
+
8
s
+
8
t
-
10
2
,
i
f
t
≠
1
26
s
-
2
2
,
i
f
t
=
1
.Proof.
Consider Graphene having “t” rows with “s” Benzene rings in each row. Let m
i
,
j denote the number of edges connecting the vertices of degrees d
i and d
j. Two-dimensional structure of Graphene (Figure 1) contains only m
2,2, m
2,3, and m
3,3 edges. The number of m
2,2, m
2,3, and m
3,3 edge in each row is mentioned in Table 1.
Therefore, Graphene containsm
2,2
=
(
t
+
4
) edges, m
2,3
=
(
4
s
+
2
t
-
4
) edges, and m
3,3
=
(
3
t
s
-
2
s
-
t
-
1
) edges.(7)
SK
G
=
∑
u
,
v
∈
E
G
d
G
u
+
d
G
v
2
,
SK
G
=
m
2,2
2
+
2
2
+
m
2,3
2
+
3
2
+
m
3,3
3
+
3
2
=
t
+
4
4
2
+
4
s
+
2
t
-
4
5
2
+
3
t
s
-
2
s
-
1
6
2
=
2
t
+
4
+
4
s
+
2
t
-
4
5
2
+
3
3
t
s
-
2
s
-
t
-
1
=
2
t
+
8
+
4
s
+
2
t
-
4
5
2
+
9
t
s
-
6
s
-
3
t
-
3
=
4
t
+
16
+
20
s
+
10
t
-
20
+
18
t
s
-
12
s
-
6
t
-
6
2
=
18
t
s
+
8
s
+
8
t
-
10
2
.Now consider the following cases.
Case
1. The SK index of Graphene for t
≠
1 is(8)
SK
G
=
18
t
s
+
8
s
+
8
t
-
10
2
.
Case
2. t
=
1, m
2,2
=
t
+
4, m
2,3
=
4
s
-
2, and m
3,3
=
s
-
2, edges as shown in Figure 2:(9)
SK
G
=
∑
u
,
v
∈
E
G
d
G
u
+
d
G
v
2
,
SK
G
=
m
2,2
2
+
2
2
+
m
2,3
2
+
3
2
+
m
3,3
3
+
3
2
=
t
+
4
4
2
+
4
s
+
2
t
-
4
5
2
+
3
t
s
-
2
s
-
t
-
1
6
2
.
Fort
=
1,(10)
=
5
2
+
4
s
-
2
5
2
+
s
-
2
3
=
10
+
4
s
-
2
5
2
+
3
s
-
6
=
20
+
20
s
-
10
+
6
s
-
12
2
=
26
s
-
2
2
.Theorem 7.
TheS
K
1 index of Graphene having “t” rows of Benzene rings with “s” Benzene rings in each row is given by(11)
S
K
1
G
=
27
t
s
+
7
t
+
6
s
-
17
2
,
i
f
t
≠
1
33
s
-
10
2
,
i
f
t
=
1
.Proof.
Consider Graphene having “t” rows with “s” Benzene rings in each row. Let m
i
,
j denote the number of edges connecting the vertices of degrees d
i and d
j. Two-dimensional structure of Graphene (Figure 1) contains only m
2,2, m
2,3, and m
3,3 edges. The number of m
2,2, m
2,3, and m
3,3 edge in each row is mentioned in Table 1.
Therefore, Graphene containsm
2,2
=
(
t
+
4
) edges, m
2,3
=
(
4
s
+
2
t
-
4
) edges, and m
3,3
=
(
3
t
s
-
2
s
-
t
-
1
) edges.(12)
S
K
1
G
=
∑
u
,
v
∈
E
G
d
G
u
·
d
G
v
2
,
S
K
1
G
=
m
2,2
2
×
2
2
+
m
2,3
2
×
3
2
+
m
3,3
3
×
3
2
=
t
+
4
4
2
+
4
s
+
2
t
-
4
6
2
+
3
t
s
-
2
s
-
1
9
2
=
2
t
+
4
+
4
s
+
2
t
-
4
3
+
3
t
s
-
2
s
-
t
-
1
9
2
=
2
t
+
8
+
12
s
+
6
t
-
12
+
3
t
s
-
2
s
-
t
-
1
9
2
=
4
t
+
16
+
24
s
+
12
t
-
24
+
27
t
s
-
18
s
-
9
t
-
9
2
=
27
t
s
+
7
t
+
6
s
-
17
2
.Now consider the following cases.
Case
1. The SK1 index of Graphene for t
≠
1 is(13)
S
K
1
G
=
27
t
s
+
7
t
+
6
s
-
17
2
.
Case
2. t
=
1, m
2,2
=
t
+
4, m
2,3
=
4
s
-
2, and m
3
,
3
=
s
-
2, edges as shown in Figure 2:(14)
S
K
1
G
=
∑
u
,
v
∈
E
G
d
G
u
·
d
G
v
2
,
S
K
1
G
=
m
2,2
2
×
2
2
+
m
2,3
2
×
3
2
+
m
3,3
3
×
3
2
=
t
+
4
4
2
+
4
s
+
2
t
-
4
6
2
+
3
t
s
-
2
s
-
1
9
2
=
2
t
+
4
+
4
s
+
2
t
-
4
3
+
3
t
s
-
2
s
-
t
-
1
9
2
.
Fort
=
1,(15)
=
2
1
+
4
+
4
s
-
2
3
+
s
-
2
9
2
=
10
+
12
s
-
6
+
s
-
2
9
2
=
20
+
24
s
-
12
+
9
s
-
18
2
=
33
s
-
10
2
.Theorem 8.
TheS
K
2 index of Graphene having “t” rows of Benzene rings with “s” Benzene rings in each row is given by(16)
S
K
2
G
=
108
t
s
+
30
t
+
28
s
-
72
4
,
i
f
t
≠
1
136
s
-
42
4
,
i
f
t
=
1
.Proof.
Consider Graphene having “t” rows with “s” Benzene rings in each row. Let m
i
,
j denote the number of edges connecting the vertices of degrees d
i and d
j. Two-dimensional structure of Graphene (Figure 1) contains only m
2,2, m
2,3, and m
3,3 edges. The number of m
2,2, m
2,3, and m
3,3 edge in each row is mentioned in Table 1.
Therefore, Graphene containsm
2,2
=
(
t
+
4
) edges, m
2,3
=
(
4
s
+
2
t
-
4
) edges, and m
3,3
=
(
3
t
s
-
2
s
-
t
-
1
) edges.(17)
S
K
2
G
=
∑
u
,
v
∈
E
G
d
G
u
+
d
G
v
2
2
,
S
K
2
G
=
m
2,2
2
+
2
2
2
+
m
2,3
2
+
3
2
2
+
m
3,3
3
+
3
2
2
=
t
+
4
4
2
2
+
4
s
+
2
t
-
4
5
2
2
+
3
t
s
-
2
s
-
1
6
2
2
=
4
t
+
4
+
4
s
+
2
t
-
4
25
4
+
9
3
t
s
-
2
s
-
t
-
1
=
4
t
+
16
+
4
s
+
2
t
-
4
25
4
+
27
t
s
-
18
s
-
9
t
-
9
=
16
t
+
64
+
100
s
+
50
t
-
100
+
108
t
s
-
72
s
-
36
t
-
36
4
=
108
t
s
+
30
t
+
28
s
-
72
4
.Now consider the following cases.
Case
1. The SK2 index of Graphene for t
≠
1 is(18)
S
K
2
G
=
108
t
s
+
30
t
+
28
s
-
72
4
.
Case
2. t
=
1, m
2
,
2
=
t
+
4, m
2,3
=
4
s
-
2, and m
3
,
3
=
s
-
2, edges as shown in Figure 2:(19)
S
K
2
G
=
∑
u
,
v
∈
E
G
d
G
u
+
d
G
v
2
2
,
S
K
2
G
=
m
2,2
2
+
2
2
2
+
m
2,3
2
+
3
2
2
+
m
3,3
3
+
3
2
2
=
t
+
4
4
2
2
+
4
s
+
2
t
-
4
5
2
2
+
3
t
s
-
2
s
-
1
6
2
2
=
4
t
+
4
+
4
s
+
2
t
-
4
25
4
+
9
3
t
s
-
2
s
-
t
-
1
.
Fort
=
1,(20)
=
4
1
+
4
+
4
s
-
2
25
4
+
9
s
-
2
=
20
+
4
s
-
2
25
4
+
9
s
-
18
=
80
+
100
s
-
50
+
36
s
-
72
4
=
136
s
-
42
4
.


### 3.1. Conclusion

A generalized formula for Arithmetic-Geometric index (AG1 index), SK index, SK1 index, and SK2 index of Graphene has been obtained without using computer.


## 3.1. Conclusion

A generalized formula for Arithmetic-Geometric index (AG1 index), SK index, SK1 index, and SK2 index of Graphene has been obtained without using computer.



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*Source: 4341919-2016-09-26.xml*