Study of Topological indices of NAn
m
and NCn
m Nanotube
October 5, 2021
S.Rajeswari1 and N. Parvathi2
1Dept.of Mathematics, SRM Institute of Science and Technology,
Kattankulathur-603 203,Tamil Nadu, INDIA
suraj1719cyr@gmail.com
2Dept.of Mathematics, SRM Institute of Science and Technology,
Kattankulathur-603 203,Tamil Nadu, INDIA
parvathn@srmist.edu.in
Abstract
A numerical quantity of a molecular graph that characterize is
topology and graph invariant is known as topological indices. The
concept of First and Second Zagreb M1(G) and M2(G). Redefined
First and Second Zagreb index ReZG1(G) and ReZG2(G) HyperZagreb index HM(G) and and Harmonic index H(G) of topological
indices of degree base vertex. In this article, M1(G) and M2(G),
ReZG1(G) and ReZG2(G), HM(G), H(G) indices are computed for
the nanotube NAn
m and NCn
m.
AMS Subject Classification: ...
Key Words and Phrases: Topological indices, First and Second
Zagreb index, Redefine First and Second Zagreb, Hyper-Zagreb index,
Harmonic index and Carbon Nanotube network..
1 Introduction
Mathematical chemistry is a branch of chemistry research in which mathematical methods are utilized to tackle problems in the field. To model
1
molecules and molecular compounds, chemical graphs are frequently used. A
molecular graph is a graph with atoms and bonds indicating the vertices and
edges, respectively. A “molecular graph” is described as a carbon molecule
with a valence of four and a maximum degree of four. [3].
A topological index is a numerical parameter that depicts the topology of a
graph. Topological indices, which are integers related to molecular graphs
of chemical compounds, are used in quantitative structure-property relationships (QSPR). The most well-known invariants of this type are degree-based
topological indices. Many aspects of chemical compounds and Nanomaterial’s, like boiling temperatures, strain energy, stiffness, and fracture toughness. Topological indices, which are invariant up to graph isomorphism, is
used to predict.
While the topology of a molecule, as represented by the molecular graph, is
essentially a non-numerical mathematical object, the measurable properties
of molecules are frequently molecular topology. The information contained
in the molecular graph must first be converted into a numerical characteristic
before the information contained in the molecular graph can be converted. A
graph invariant is an integer that can only be determined by a graph. These
invariants are popularly known as “Topological Indices.”
Nanotechnology is the study of nanostructures of sizes 1 to 100 nanometres. Sumio Iijima, a Japanese scientist, discovered the carbon nanotube in
1991[2]. (CNTs) are initially used it as an additive in a variety of structural materials for electronics, plastics, and other nanotechnology products.
Carbon nanotubes are entirely made up of carbon arranged in a tubular
structure made up of condensed benzene rings. Allotropes of carbon with a
cylindrical nanostructure make carbon nanotubes. (CNTs) are of two types:
Single Walled Carbon Nanotube (SWCNT) or multiple sheets of graphene
Multi Walled Carbon Nanotube (MWCNT) (MWCNT). Carbon nanotubes
and fullerenes are hollow carbon allotropes with exceptional thermal, electrical, and mechanical. Buckyballs are spherical fullerenes, and nanotubes are
cylindrical fullerenes. The walls of these structures are made of a single layer
of carbon atoms (graphene).
These cylindrical carbon particles have distinct properties that are useful in
nanotechnology, optics, electronics, and a broad range of materials research
and innovation disciplines. [1-3]. Carbon nanotubes are the stiffest and most
durable materials. Carbon nanotubes are well suited for manipulating other
Nano scale structures due to their flexibility and strength, implying that they
will play an important role in nanotechnology engineering.
2
Nanotubes are good heat conductors and are a good product for electronics and optical. Nanotubes are being developed for a variety of purposes.
High-quality nanotube materials are sought for both basic and technological
applications. The absence of structural and chemical flaws at a large length
scale (e.g. 1-10 microns) along the tube axes is referred to as high quality. Several approaches have been devised to generate CNTs and MWNTs
in laboratory quantities with various structures and morphologies. Carbon
nanotube produce three methods: Arc discharge, Laser ablation, and chemical vapour deposition [16] [17].
The discovery of fullerene molecules and related carbon structures such as
nanotubes has sparked a surge in chemistry, physics, and materials research.
The atoms are organized on a pseudo-spherical framework composed primarily of pentagons and hexagons. Its chemical graph consists of only hexagonal
and (exactly 12) pentagonal faces embedded on the surface of a sphere [19].
Let G be a connected graph with V (G) and E(G) as he vertex set and the
edge set, respectively. The distance between two vertices says a and b of
G, denoted by d(a, b) is defined as the number of edges in the shortest path
connecting a and b.
2 Definitions and Literature Review
Let G be a connected graph with V (G) and E(G) as he vertex set and the
edge set, respectively. The distance between two vertices says u and v of
G, denoted by d(u, v) is defined as the number of edges in the shortest path
connecting u and v.
Definition 1. An important topological index introduced more than 30 years
ago by I. Gutman and N. Trinajistic is the first and second Zagreb indices
(M1(G) and M2(G)) [3]. These indices are defined as:
M1(G) = X
e=uv∈E(G)
(du + dv)
M2(G) = X
e=uv∈E(G)
(du.dv)
Definition 2. The Hyper Zagreb index (HM(G)) being introduced by Shirdel
3
et al. It is a distance based Zagreb index in 1913 [9].
HM(G) = X
e=uv∈E(G)
(du + dv)
2
Definition 3. The Sum Connectivity Index (SCI(G)) being introduced by
Zhou and Trinajstic[11].
SCI(G) = X
e=uv∈E(G)
1
√
du + dv
Definition 4. The Redefined First and Second Zagreb index being introduced by Rajini et. al. For a graph G and there are manifested as
Re ZG1(G) = X
e=uv∈E(G)
d(u) + d(v)
d(u).d(v)
Re ZG2(G) = X
e=uv∈E(G)
d(u).d(v)
d(u) + d(v)
Definition 5. For a graph G, the Harmonic index is defined as
H(G) = X
e=ab∈E(G)
2
du + dv
3 Main Results and Discussion
We will take a look at the carbon nanotube network in this section. Consider
the mn quadrilateral section P
n
m, which is cut from the regular hexagonal
lattice L and has m ≥ 2 hexagons on the top and bottom sides and n ≥ 2
hexagons on the lateral sides (see figure).
If we identify two lateral sides such that the vertices u
j
0
and u
j
m for j =
0, 1, 2, · · · , n are identified, we get the nanotube NAn
m with 2m(n+1) vertices
and (3n + 2)m edges.
Let n be an even number, n ≥ 2 and m ≥ 2. If we identify the top and
bottom sides of the quadrilateral section P
n
m in such a way that we identify
the vertices u
0
i
and u
n
i
for i = 0, 1, 2, · · · , m and the vertices v
0
i
and v
n
i
for
i = 0, 1, 2, · · · , m, we get the nanotube NCn
m of order n(2m + 1) and size
n(3m + 1).
4
Figure 1: Quadrilateral section P
n
m cuts from the regular hexagonal lattice
Theorem 6. Consider the nanotube NAn
m for m, n ≥ 2. Then
M1(NAn
m) = 18mn + 18m
M2(NAn
m) = 27mn + 6m
Proof:
The nanotube NAn
m has 2m vertices of degree 2 and 2mn vertices of degree
3. The edge set E(NAn
m) divides into two edge partitions based on degrees
of end vertices.
The first edge partition E1(NAn
m) contains m (3n-2) edges, where deg(a) =
deg(b) = 3.
The second edge partition E2(NAn
m) contains 4m edges, where deg(a) = 2
and deg(b) = 3.
By definition of First Zagreb index, we have
M1(G) = X
e=uv∈E(G)
(du + dv)
5
It follows that
M1(NAn
m) = X
e=uv∈E1(NAnm)
(du + dv)
= 6|E1(NAn
m)| + 5|E2(NAn
m)|
= 6m(3n − 2) + 5(4m)
= 18mn − 12m + 20m
= 18mn + 8m
Similarly, definition Second Zagreb index, we have
M2(G) = X
e=uv∈E(G)
(du.dv)
It follows that
M2(NAn
m) = X
e=uv∈E2(NAnm)
(du.dv)
= 9|E1(NAn
m)| + 6|E2(NAn
m)|
= 9m(3n − 2) + 6(4m)
= 27mn − 18m + 24m
= 27mn + 6m
Hence proved.
Theorem 7. Consider the nanotube NCn
m for m ≥ 2, n ≥ 2. Then
M1(NCn
m) = 18mn − n
M2(NCn
m) = 54mn − 13n
2
Proof:
Let the nanotube NCn
m has 2n vertices of degree 2 and n (2m-1) vertices of
degree 3.
Here are three types of edges in E(NCn
m) based on the degrees of end vertices
of each edge.
(i.e.) E(NCn
m) = E1(NCn
m) ∪ E2(NCn
m) ∪ E2(NCn
m).
The edge partition E1(NCn
m) contains n edges where deg(a) = deg(b) = 2.
6
The edge partition E2(NCn
m) contains 2n edges where deg(a) = 2 and deg(b) =
3.
The edge partition E3(NCn
m) contains n(3m − 5/2) edges where deg(a) =
deg(b) = 3.
By definition of Firs Zagreb index, we have
M1(G) = X
e=uv∈E(G)
(du + dv)
It follows that
M1(NAn
m) = X
e=uv∈E1(NCnm)
(du + dv)
= (2 + 2)|E1(NCn
m)| + (2 + 3)|E2(NCn
m)| + (3 + 3)|E3(NCn
m)|
= 4(n) + 5(2n) + 6[n(3m − 5/2)]
= 4n + 10n + 18mn − 15n
= 18mn − n
Also by definition of Second Zagreb index, we have
M2(NCn
m) = X
e=uv∈E(NCnm)
(du.dv)
It follows that
M2(NAn
m) = X
e=uv∈E2(NCnm)
(du.dv)
= (2 × 2)|E1(NCn
m)| + (2 × 3)|E2(NCn
m)| + (3 × 3)|E3(NCn
m)|
= 4(n) + 6(2n) + 9[n(3m − 5/2)]
= 4n + 12n + 27mn −
45n
2
=
54mn − 13n
2
Hence the proof.
Theorem 8. Consider the nanotube NAn
m for m, n ≥ 2 then the Hyper
Zagreb index is HM(NAn
m) = 108mn − 28m where m, n ≥ 2.
7
Proof:
Let the nanotube NAn
m has 2m vertices of degree 2 and 2mn vertices of
degree 3.
For determining the Hyper-Zagreb index of nanotube NAn
m we apply the
edge partitions described in the proof of the theorem 6. Then we have
HM(G) = X
e=uv∈E(G)
(du + dv)
2
We obtain
HM(NAn
m) = (3 + 3)2
|E1(NAn
m)| + (2 + 3)2
|E2(NAn
m)|
= 36[m(3n − 2)] + 25(4m)
= 108mn − 72m + 100m
= 108mn − 28m
Hence the proof.
Theorem 9. Consider the nanotube NCn
m for m ≥ 2 and n ≥ 2. Then
the Hyper Zagreb index is HM(NCn
m) = 108mn + 21n.
Proof:
Let the nanotube NCn
m has 2n vertices of degree 2 and n(2m − 1) vertices of
degree 3.
The edge set can be divided into three types of edges based on degrees of
end vertices of each edge.
For computing, the Hyper-Zagreb index of the nanotube NCn
m, we apply the
edge values are defined by the proof of the theorem 7.
By definition of Hyper-Zagreb index
HM(G) = X
e=uv∈E(G)
(du + dv)
2
8
Then we have
HM(G) = X
e=uv∈E(NCnm)
(du + dv)
2
= (2.2)2
|E1(NCn
m)| + (2.3)2
|E2(NCn
m)| + (3.3)2
|E3(NCn
m)|
= 16(n) + 25(2n) + 81[n(3m − 5/2)]
= 16n + 50n + 243mn − 405n/2
=
1
2
[243mn − 273n]
Hence the proof.
Theorem 10. Consider the nanotube NAn
m for m, n ≥ 2 then the
Harmonic index is H(G) = 1
3
[3mn + 2m]
Let us consider the nanotube NAn
m listed in the proof theorem 6 we
obtain.
H(G) = X
e=uv∈E(G)
2
du + dv
=
2
3 + 3
|E1(NAn
m)| +
2
2 + 3
|E2(NAn
m)|
=
1
3
[m(3n − 2)] + 2
5
(4m)
= mn −
2
3
m +
8
5
m
=
15mn − 10m + 24m
15
H(NAn
m) = 1
3
[3mn + 2m]
Hence the proof.
Theorem 11. Consider the nanotube NCn
m for m, n ≥ 2 then the
Harmonic index is H(G) = 1
30
[30mn + 14n]
Proof:
Let the nanotube NCn
m has 2n vertices of degree 2 and n(2m − 1) vertices of
9
degree 3.
For determining the Harmonic index of nanotube NCn
m we apply the edge
partitions described in the proof of the theorem 7. Then we have
By definition of Harmonic index
H(G) = X
e=uv∈E(G)
2
du + dv
=
2
2 + 2
|E1(NCn
m)| +
2
2 + 3
|E2(NCn
m)| +
2
3 + 3
|E3(NCn
m)|
=
1
2
(n) + 2
5
(2n) + 1
3
[n(3m − 5/2)]
=
n
2
+
4n
5
+ mn −
5n
6
=
30mn + 15n + 24n − 25n
30
=
1
30
[30mn + 14n]
Hence the proof.
Theorem 12. Consider the nanotube NAn
m for m, n ≥ 2. Then the
Sum Connectivity index is
SCI(G) = 1
√
6
[3mn − 2m] + 1
√
5
(4m)
Proof:
Let the nanotube NAn
m has 2m vertices of degree 2 and 2mn vertices of
degree 3.
The edge set E(NAn
m) divides into two edge partitions based on degrees of
end vertices.
The first edge partition E1(NAn
m) contains m(3n − 2) edges, where deg(a) =
deg(b) = 3.
The second edge partition E2(NAn
m) contains 4m edges, where deg(a) = 2
and deg(b) = 3.
By definition of Sum Connectivity index, we have
SCI(G) = X
e=uv∈E(G)
1
√
du + dv
10
It follows that
SCI(NAn
m) = X
e=uv∈E(NAnm)
1
√
du + dv
=
1
√
6
|E1(NAn
m)| +
1
√
5
|E2(NAn
m)|
=
1
√
6
[m(3n − 2)] + 1
√
5
(4m)
=
1
√
6
[3mn − 2m] + 1
√
5
(4m)
Hence the proof.
Theorem 13. Consider the nanotube NCn
m for m, n ≥ 2. Then the
Sum Connectivity index is
SCI(NCn
m) = n
2
+
2n
√
5
+
1
√
6
6mn − 5n
2
Proof:
Let us consider the nanotube NCn
m has 2n vertices of degree 2 and n(2m−1)
vertices of degree 3.
The edge set can be divided into three types of edges based on degrees of
end vertices of each edge.
The edge set E1(NCn
m) contains n edges where deg(u) = deg(v) = 2.
The edge set E2(NCn
m) contains 2n edges where deg(u) = 2 and deg(v) = 3.
The edge set E3(NCn
m) contains n(3m−5/2) edges where deg(u) = deg(v) =
3.
By definition of Sum Connectivity index
SCI(G) = X
e=uv∈E(G)
1
√
du + dv
11
It follows that
SCI(G) = X
e=uv∈E(NCnm)
1
√
du + dv
=
1
2
|E1(NCn
m)| +
1
√
5
|E2(NCn
m)| +
1
√
6
|E3(NCn
m)|
=
1
2
(n) + 1
√
5
(2n) + 1
√
6
[n(3m − 5/2)]
=
n
2
+
2n
√
5
+
1
√
6
6mn − 5n
2
Hence the proof.
4 Conclusion
In this article, we determined the degree-based topological indices. We compute the First and Second Zagreb index, Hyper - Zagreb index, Harmonic
index, and Sum Connectivity index for the nanotube NAn
m and NCn
m.
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