Hindawi
Journal of Chemistry
Volume 2021, Article ID 3734185, 15 pages
https://doi.org/10.1155/2021/3734185
Research Article
Novel Degree-Based Topological Descriptors of
Carbon Nanotubes
M. C. Shanmukha,1 A. Usha,2 M. K. Siddiqui,3 K. C. Shilpa,4 and A. Asare-Tuah 5
1Department of Mathematics, Jain Institute of Technology, Davanagere-577003, Karnataka, India
2Department of Mathematics, Alliance School of Applied Mathematics, Alliance University, Bangalore-562106, Karnataka, India
3Department of Mathematics, Comsats University Islamabad, Lahore Campus, Lahore, Pakistan
4Department of Computer Science and Engineering, Bapuji Institute of Engineering and Technology, Davanagere-577004,
Karnataka, India
5Department of Mathematics, University of Ghana, Legon, Ghana
Correspondence should be addressed to A. Asare-Tuah; aasare-tuah@ug.edu.gh
Received 27 July 2021; Accepted 27 August 2021; Published 8 September 2021
Academic Editor: Ajaya Kumar Singh
Copyright © 2021 M. C. Shanmukha et al. )is is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is
properly cited.
)e most significant tool of mathematical chemistry is the numerical descriptor called topological index. Topological indices are
extensively used in modelling of chemical compounds to analyse the studies on quantitative structure activity/property/toxicity
relationships and combinatorial library virtual screening. In this work, an attempt is made in defining three novel descriptors,
namely, neighborhood geometric-harmonic, harmonic-geometric, and neighborhood harmonic-geometric indices. Also, the
aforementioned three indices along with the geometric-harmonic index are tested for physicochemical properties of octane
isomers using linear regression models and computed for some carbon nanotubes.
1. Introduction compounds where, generally, the hydrogen atoms are
suppressed. )e originality of QSAR/QSPR models depends
)e applications of graph theory are diversified in every field, on physicochemical properties for chemical compounds
but chemistry is the major area of the implementation of with high degree of precision. )ese models depend on
graph theory. In chemical graph theory, topological index various factors such as selecting the suitable compounds,
plays a vital role which facilitates the chemists with a treasure suitable descriptors, and suitable algorithms or tools used in
of data that correlate with the structure of the chemical model development [12]. )e QSAR/QSPR analysis is based
compound. )e topological index is a numerical descriptor, on the data obtained by the numerical descriptors. )ese
defines the graph topology of the molecule, and predicts an data are used to verify whether the compound under the
extensive range of molecular properties 5[1–6]. study is suitable for drug making as the TIs provide com-
From the last two decades, topological indices (TIs) are putational data about the compound. Considering the in-
identified and used in pharmacological medicine, bio- formation of the compound, QSAR/QSPR/QSTR analyses
inorganic chemistry, toxicity, and theoretical chemistry and are carried out.
are also used for correlation analysis [7–11]. )e TIs have increasing popularity in the field of re-
Topological descriptors are frequently used in the dis- search as they involve only computation without performing
covery of drugs as they have rich datasets that give high any physical experiment. Recent years have proved con-
predictive values. )ese descriptors give the information siderable attention in TIs as the effects of an atomic type and
depending on the arrangement of atoms and their bonds of a group efforts are considered in QSAR/QSPR modelling
chemical compound. )ey are studied for chemical [13–15]. Distance-based TIs are used in QSAR analysis,
2 Journal of Chemistry
while chirality descriptors are introduced based on molec- placed next to each other. )is gives two different types of
ular graphs [16]. configurations with different terminologies discussed now.
Alkanes are acyclic saturated hydrocarbons in which To explain the structure of a nanotube that is infinitely long,
carbons and hydrogens are arranged in a tree-like structure. we imagine it to be cut open by a parallel axis and placed on a
)e main use of alkanes is found in crude oil such as pe- plane. )en, the atoms and bonds coincide with an imag-
troleum, cooking gas, pesticides, and drug synthesis. )e inary graphene sheet. )e length of the two atoms on op-
compounds that contain absolutely the same number of posite edges of the strip corresponds to circumference of the
atoms but their arrangement differs are termed as isomers. A cylindrical graphene sheet [27–29].
study is carried out for eighteen octane isomers (refer )e main objectives of this work are as follows:
Figure 1).
A structure whose size is between the microscopic and To define novel indices
molecular structure is referred to as a nanostructure. )ere To discuss the physical and chemical applicability of
are different types of nanostructures, namely, nanocages, octane isomers using regression models
nanocomposites, nanoparticles, nanofabrics, etc. In the re- To compute defined indices for carbon nanotubes such
cent years, nanostructures have attracted a lot of researchers as C4C8(S), C4C8(R), and H-naphthalenic nanosheets
in the areas of biology, chemistry, and medicines. Topo-
logical indices of nanostructures can be studied from Let G � (V, E) be a graph with a vertex set V(G) and an
[17–24]. )e nanostructures made of carbons with cylin- edge set E(G) such that |V(G)| � n and |E(G)| � m. For
drical shape are carbon nanotubes (CNTs). )ey have a standard graph terminologies and notations, refer to
similar structure to that of a fullerene and graphene except [30, 31].where (u, v) is an element of E(G), du represents the
their cylindrical shape. )e shape of fullerene is as that of a degree of the vertex u, and Su represents the neighborhood
football or basketball design where hexagons are connected. degree of the vertex u.
In 1991, Iijima [25] used carbon nanotubes that have
attracted many researchers in nanoscience and nanotech- Definition 1. Recently, Usha et al. [32] defined the geo-
nology worldwide. As they have exotic properties, they are metric-harmonic (GH) index, inspired by Vukicevic and
widely used in both research and applications. Nanotubes Furtula [33] in designing the GA index:
have a distinctive structure with remarkable mechanical and 􏽰�����( d + d 􏼁 d · d
electrical properties. In case of carbon nanotubes, the GH(G) � 􏽘 u v u v. (1)
hexagons are surrounded by squares, and each of these 2
patterns is linearly arranged. Carbon nanotubes reveal ex- Motivated by the above work, in this paper, an attempt is
ceptional electrical conductivity and possess wonderful made to define three novel indices based on degree and
tensile strength and thermal conductivity as they have neighborhood degree, namely, harmonic-geometric (HG),
nanostructures in which the carbon atoms are strongly neighborhood geometric-harmonic (NGH), and neighbor-
connected. hood harmonic-geometric (NHG) indices. )ey are defined
Carbon nanotubes have applications in orthopaedic as follows:
implants, especially in total hip replacement and other 2
treatments pertaining to bone-related ailments. )ey are HG(G) � 􏽘 􏽰����� ,
used as a grouting agent placed between the prosthesis and ( du + dv􏼁 du · dv
the bone as a part of their therapeutic use.)e CNTs are used 􏽰�����
in biomedical fields because of their structural stiffness and ( Su + SNGH v
􏼁 Su · Sv
(G) � 􏽘 , (2)
effective optical absorption from UV to IR. Also, they can be 2
altered chemically which are expected to be useful in many 2
fields of technology such as electronics, composited mate- NHG(G) � 􏽘 􏽰�����,
rials, and carbon fibres. )ey have incredible applications in ( Su + Sv􏼁 Su · Sv
the field of materials science [26]. When the hexagonal
lattice is rolled in different directions, it looks like single-wall
carbon nanotubes have spiral shape and translational
symmetry along the tube axis. It has rotational symmetry 1.1. Chemical Applicability of GH, NGH, HG, and NHG
along its own axis. Even though nanotubes have favourable Indices. In this section, a linear regression model of four
applications in a variety of fields, their large-scale production physical properties is presented for GH, NGH, HG, and
has been restricted. )e main constraint that obstructs their NHG indices. )e physical properties such as entropy(S),
use lies in difficulty in controlling their structure, impurities, acentric factor (AF), enthalpy of vaporization (HVAP), and
and poor process ability. To enhance their usage, they have standard enthalpy of vaporization (DHVAP) of octane
grabbed the attention especially in the formation of com- isomers have shown good correlation with the indices
posites with polymers. considered in the study. )e GH, HG, NGH, and NHG
)ere are two types of configurations in the arrangement indices are tested for the octane isomers’ database available
of nanotubes, namely, zigzag and armchair. In the zigzag at https://www.moleculardescriptors.eu/dataset.htm. )e
configuration, the hexagons are placed one below the other GH, HG, NGH, and NHG indices are computed and tab-
linearly, whereas in the armchair configuration, they are ulated in columns 6, 7, 8, and 9 of Table 1.
Journal of Chemistry 3
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
(j) (k) (l)
(m) (n) (o)
(p) (q) (r)
Figure 1: (a) n-Octane, (b) 2-methylheptane, (c) 3-methylheptane, (d) 4-methylheptane, (e) 3-ethylhexane, (f ) 2,2-dimethylhexane, (g) 2,3-
dimethylhexane, (h) 2,4-dimethylhexane, (i) 2,5-dimethylhexane, (j) 3,3-dimethylhexane, (k) 3,4-dimethylhexane, (l) 3-ethyl-2-methyl-
pentane, (m) 3-ethyl-3-methylpentane, (n) 2,2,3-trimethylpentane, (o) 2,2,4-trimethylpentane, (p) 2,3,3-trimethylpentane, (q) 2,3,4-tri-
methylpentane, and (r) 2,2,3,3-trimethylbutane.
Table 1: Experimental values of S, AF, HVAP, and DHVAP and the corresponding values of the GH index, HG index, NGH index, and
NHG index of octane isomers.
Alkane S AF HVAP DHVAP GH NGH HG NHG
n-Octane 111.700 0.398 73.190 9.915 24.423 84.496 2.193 0.679
2-Methylheptane 109.800 0.378 70.300 9.484 27.173 98.746 1.962 0.573
3-Methylheptane 111.300 0.371 71.300 9.521 27.954 107.475 2.058 0.568
4-Methylheptane 109.300 0.372 70.910 9.483 27.954 113.356 2.058 0.600
3-Ethylhexane 109.400 0.362 71.700 9.476 28.735 131.941 2.154 0.476
2,2-Dimethylhexane 103.400 0.339 67.700 8.915 33.607 122.970 1.689 0.471
2,3-Dimethylhexane 108.000 0.348 70.200 9.272 31.637 110.622 1.862 0.498
2,4-Dimethylhexane 107.000 0.344 68.500 9.029 30.885 123.236 1.827 0.474
2,5-Dimethylhexane 105.700 0.357 68.600 9.051 32.248 133.242 1.731 0.469
3,3-Dimethylhexane 104.700 0.323 68.500 8.973 35.213 151.398 1.829 0.449
3,4-Dimethylhexane 106.600 0.340 70.200 9.316 32.418 117.701 1.958 0.578
2-Methyl-3-ethylpentane 106.100 0.332 69.700 9.209 32.418 176.111 1.958 0.342
3-Methyl-3-ethylpentane 101.500 0.307 69.300 9.081 36.820 149.222 1.968 0.377
2,2,3-Trimethylpentane 101.300 0.301 67.300 8.826 38.834 164.366 1.606 0.338
2,2,4-Trimethylpentane 104.100 0.305 64.870 8.402 36.537 155.874 1.459 0.358
2,3,3-Trimethylpentane 102.100 0.293 68.100 8.897 39.659 141.657 1.649 0.462
2,3,4-Trimethylpentane 102.400 0.317 68.370 9.014 35.321 168.797 1.666 0.390
2,2,3,3-Trimethylbutane 93.060 0.255 66.200 8.410 46.000 223.620 1.263 0.227
4 Journal of Chemistry
Using the method of least squares, the linear regression From Table 2 and Figure 2, it is obvious that the GH
models for S, AF, HVAP, and DHVAP are fitted using the index highly correlates with the acentric factor and the
data of Table 1. correlation coefficient |r| � 0.987. Also, the GH index has
)e fitted models for the GH index are good correlation coefficient |r| � 0.968 with entropy, |r| �
0.815 with HVAP, and |r| � 0.885 with DHVAP.
S � 133.078(± 1.82) − 0.833(± 0.054)GH, (3) From Table 3 and Figure 3, it is noticed that the HG
index highly correlates with DHVAP and the correlation
acentric factor � 0.557(± 0.009) − 0.007(± 0.000)GH, coefficient r � 0.939. Also, the HG index has good corre-
4 lation coefficient r � 0.85 with entropy, r � 0.833 with the( ) acentric factor, and r � 0.935 with HVAP.
HVAP 79 613 ± 1 878 0 315 ± 0 056 GH 5 From Table 4 and Figure 4, it is clear that the NGH index� . ( . ) − . ( . ) , ( ) highly correlates with the acentric factor and the correlation
coefficient |r| � 0.877. Also, the NGH index has good cor-
DHVAP � 11.273(± 0.285) − 0.065(± 0.008)GH. (6) relation coefficient |r| � 0.873 with entropy, |r| � 0.695 with
)e fitted models for the HG index are HVAP, and |r| � 0.778 with DHVAP.From Table 5 and Figure 5, it is clear that the NHG index
76 608 ± 4 486 15 766 ± 2 435 HG 7 highly correlates with the acentric factor and the correlationS � . ( . ) + . ( . ) , ( )
coefficient r � 0.877. Also, the NHG index has good cor-
relation coefficient r � 0.852 with entropy, r � 0.777 with
acentric factor � 0.114(± 0.037) + 0.121(± 0.020)HG, HVAP, and r � 0.843 with DHVAP.
(8)
HVAP 54 960 ± 1 356 7 773 ± 0 736 HG 9 2. GH, NGH, HG, and NHG Indices of C C (S),� . ( . ) + . ( . ) , ( ) 4 8
C4C8(R), and H-Naphthalenic Nanosheets
DHVAP � 6.428(± 0.249) + 1.477(± 0.135)HG. (10) 2.1. Results for the C4C8(S) Nanosheet. )e alternating
)e fitted models for the NGH index are pattern of 4 carbon atoms forming squares and 8 carbonatoms forming octagons constitutes the TUC4C8(S)[a, b]
nanosheet.
S � 121.77(± 2.35) − 0.119(± 0.017)NGH, (11)
In this section, GH, HG, NGH, and NHG indices of the
C4C8(S) nanosheet are computed. )e pattern of carbonacentric factor � 0.465(± 0.018) − 0.001(± 0.000)NGH, atoms gives rise to two types of nanosheets, namely,
(12) T1[a, b] and T2[a, b]. )e 2-dimensional nanosheet is
represented by T1[a, b], where a and b are parameters
HVAP � 75.007(± 1.552) − 0.043(± 0.011)NGH, (13) (Figure 6). In T1[a, b], C4 acts as a square, while C8 is an
octagon in which a and b represent the column and row,
DHVAP � 10.363(± 0.257) − 0.009(± 0.002)NGH. (14) respectively. Figure 7 depicts the type 1 − C4C8(S)
nanosheet. )e number of vertices of the C4C8(S)
)e fitted models for the NHG index are nanosheet is 8ab, and the number of edges is
S � 89.524 ± 2 505 34 35 ± 5 271 NHG 15 12ab − 2a − 2b.( . ) + . ( . ) , ( ) )e edge partition of the T1[a, b] nanosheet based on the
degree of vertices is detailed in Table 6.
acentric factor � 0.207(± 0.018) + 0.277(± 0.038)NHG,
(16) Theorem 1. Let T1[a, b] be an (a, b)-dimensional nano-
HVAP � 62.667(± 1.352 14 044 ± 2 846 NHG 17 sheet; then, GH and HG indices are equal to) + . ( . ) , ( )
1 4938 4938 376
DHVAP � 7.793(± 0.219) + 2.882(± 0 460 NHG 18 GH􏼐T [a, b]􏼑 � 108ab − a − b + ,. ) . ( ) 125 125 125
(19)
Note: in equations (3)–(18), the errors of the regression 1 4 33 33 69HG􏼐T [a, b]􏼑 � ab + a + b + .
coefficients are represented within brackets. 3 125 125 500
Tables 2–5 and Figures 2–5 show the correlation coef-
ficient and residual standard error for the regression models
of four physical properties with GH, HG, NGH, and NHG Proof. Using Table 6, the definitions of GH and HG indices
indices. are as follows:
Journal of Chemistry 5
Table 2: Parameters of regression models for the GH index.
Physical properties Value of the correlation coefficient Residual standard error
Entropy −0.968 1.17
Acentric factor −0.987 0.0059
HVAP −0.815 1.21
DHVAP −0.885 0.184
Table 3: Parameters of regression models for the HG index.
Physical properties Value of the correlation coefficient Residual standard error
Entropy 0.85 2.448
Acentric factor 0.833 0.020
HVAP 0.935 0.74
DHVAP 0.939 0.136
Table 4: Parameters of regression models for the NGH index.
Physical properties Value of the correlation coefficient Residual standard error
Entropy −0.873 2.274
Acentric factor −0.877 0.0176
HVAP −0.695 1.502
DHVAP −0.778 0.248
Table 5: Parameters of regression models for the NHG index.
Physical properties Value of the correlation coefficient Residual standard error
Entropy 0.852 2.436
Acentric factor 0.877 0.018
HVAP 0.777 1.315
DHVAP 0.843 0.213
115
0.40
110 0.38
0.36
105 0.34
0.32
100 0.30
95 0.28
0.26
90 0.24
25 30 35 40 45 50 25 30 35 40 45 50
GH index GH index
(a) (b)
74 10.2
10.0
72 9.8
9.6
70 9.4
9.2
68 9.0
8.8
66 8.6
8.4
64 8.2
25 30 35 40 45 50 25 30 35 40 45 50
GH index GH index
(c) (d)
Figure 2: Scatter diagram of physical properties S, AF, HVAP, and DHVAP with the GH index.
HVAP S
DHVAP AF
6 Journal of Chemistry
115
0.40
110 0.38
0.36
105 0.34
0.32
100 0.30
95 0.28
0.26
90 0.24
1.2 1.4 1.6 1.8 2.0 2.2 1.2 1.4 1.6 1.8 2.0 2.2
HG index HG index
(a) (b)
74 10.2
10.0
72 9.8
9.6
70 9.4
9.2
68 9.0
8.8
66 8.6
8.4
64 8.2
1.2 1.4 1.6 1.8 2.0 2.2 1.2 1.4 1.6 1.8 2.0 2.2
HG index HG index
(c) (d)
Figure 3: Scatter diagram of physical properties S, AF, HVAP, and DHVAP with the HG index.
115
0.40
110 0.38
0.36
105 0.34
0.32
100 0.30
95 0.28
0.26
90 0.24
80 100 120 140 160 180 200 220 240 80 100 120 140 160 180 200 220 240
NGH index NGH index
(a) (b)
74 10.2
10.0
72 9.8
9.6
70 9.4
9.2
68 9.0
8.8
66 8.6
8.4
64 8.2
80 100 120 140 160 180 200 220 240 80 100 120 140 160 180 200 220 240
NGH index NGH index
(c) (d)
Figure 4: Scatter diagram of physical properties S, AF, HVAP, and DHVAP with the NGH index.
HVAP S
HVAP S
DHVAP AF DHVAP AF
Journal of Chemistry 7
115 0.40
0.38
110 0.36
105 0.34
0.32
100 0.30
0.28
95 0.26
90 0.24
0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0.6 0.7
NHG index NHG index
(a) (b)
74 10.2
10.0
72 9.8
9.6
70 9.4
9.2
68 9.0
8.8
66 8.6
8.4
64 8.2
0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0.6 0.7
NHG index NHG index
(c) (d)
Figure 5: Scatter diagram of physical properties S, AF, HVAP, and DHVAP with the NHG index.
Figure 6: A TUC4C8(S)[a, b] nanotube.
1 2 3 a
1
2
3
b
Figure 7: Type I-C C (S) nanosheet T14 8 [a, b].
HVAP S
DHVAP AF
8 Journal of Chemistry
Table 6: )e edge partition of T1[a, b].
(du, dv) with uv ∈ E(G) Number of edges
(2, 2) 2(a + b + 2)
(2, 3) 4a + 4b − 8
(3, 3) 12ab − 8a − 8b + 4
􏽰������
1 ( d + d 􏼁 d × dGH􏼐T [a, b] � 􏽘 u v u v􏼑
uv∈ 2E(G)
√���� √����
(2 + 2)( 2 × 2 ) (2 + 3)( 2 × 3 )
� (2a + 2b + 4)􏼨 􏼩 +(4a + 4b − 8)􏼨 􏼩
2 2
√����
(2 + 3)( 2 × 3 )
+(12ab − 8a − 8b + 4)􏼨 􏼩
2
1 4938 4938 376GH􏼐T [a, b]􏼑 � 108ab − a − b + ,
125 125 125
(20)
2
HG 1􏼐T [a, b]􏼑 � 􏽘 􏽰������
∈ ( d + d 􏼁 d × duv E(G) u v u v
2 2
� (2a + 2b + 4)􏼨 √���� 􏼩 +(4a + 4b − 8)􏼨 √���� 􏼩
(2 + 2)( 2 × 2 ) (2 + 3)( 2 × 3 )
2
+(12ab − 8a − 8b + 4)􏼨 √���� 􏼩
(2 + 3)( 2 × 3 )
1 4 33 33 69HG􏼐T [a, b]􏼑 � ab + a + b + .
3 125 125 500
□
)e edge partition of the T1[a, b] nanosheet based on the Theorem 2. Let T1[a, b] be an (a, b)-dimensional nano-
neighborhood degree of vertices is detailed in Table 7. sheet; then, NGH and NHG indices are equal to
1 251531 251531 159121NGH􏼐T [a, b]􏼑 � 972ab − a − b + ,
500 500 1000
(21)
1 37 91 91 157NHG􏼐T [a, b]􏼑 � ab + a + b + .
250 1000 1000 1000
Proof. Using Table 7, the definitions of NGH and NHG
indices are as follows:
Journal of Chemistry 9
Table 7: Edge partition of T1[a, b] for neighborhood degree-based vertices.
(Su, Sv) with uv ∈ E(G) Number of edges
(4, 4) 4
(4, 5) 8
(5, 5) 2a + 2b − 8
(5, 8) 4a + 4b − 8
(8, 8) 2a + 2b − 4
(8, 9) 4a + 4b − 8
(9, 9) 12ab − 14a − 14b + 16
􏽰������
1 ( Su + Sv􏼁 Su × SNGH v􏼐T [a, b]􏼑 � 􏽘
uv∈ 2E(G)
√���� √���� √����
(4 + 4)( 4 × 4 ) (4 + 5)( 4 × 5 ) (5 + 5)( 5 × 5 )
� 4􏼨 􏼩 + 8􏼨 􏼩 +(2a + 2b − 8)􏼨 􏼩 +(4a + 4b − 8)
2 2 2
√���� √���� √����
(5 + 8)( 5 × 8 ) (8 + 8)( 8 × 8 ) (8 + 9)( 8 × 9 )
· 􏼨 􏼩 +(2a + 2b − 4)􏼨 􏼩 +(4a + 4b − 8)􏼨 􏼩
2 2 2
√����
(9 + 9)( 9 × 9 )
+(12ab − 14a − 14b + 16)􏼨 􏼩
2
1 251531 251531 159121NGH􏼐T [a, b]􏼑 � 972ab − a − b + ,
500 500 1000
􏽰������ (22)
1 ( S + S 􏼁 S × SNHG u v u v􏼐T [a, b]􏼑 � 􏽘
∈ 2uv E(G)
2 2 2
� 4􏼨 √���� 􏼩 + 8􏼨 √���� 􏼩 +(2a + 2b − 8)􏼨 √���� 􏼩 +(4a + 4b − 8)
(4 + 4)( 4 × 4 ) (4 + 5)( 4 × 5 ) (5 + 5)( 5 × 5 )
2 2 2
· 􏼨 √���� 􏼩 +(2a + 2b − 4)􏼨 √���� 􏼩 +(4a + 4b − 8)􏼨 √���� 􏼩
(5 + 8)( 5 × 8 ) (8 + 8)( 8 × 8 ) (8 + 9)( 8 × 9 )
2
+(12ab − 14a − 14b + 16)􏼨 √���� 􏼩
(9 + 9)( 9 × 9 )
NHG 1
37 91 91 157
􏼐T [a, b]􏼑 � ab + a + b + .
250 1000 1000 1000 □
2.2. Results for the C4C8(R) Nanosheet. )is structure is )e edge partition of the T2[a, b] nanosheet for degree-
formed by 4 carbon atoms forming a rhombus that are based vertices is detailed in Table 8.
linearly bridged by edges whose sequence looks like 4
rhombuses connected by 4 edges row and column wise
resulting in an alternating pattern of rhombuses and octa- Theorem 3. Let T2[a, b] be an (a, b)-dimensional nano-
gons and is represented as T2[a, b]. )e 2-dimensional sheet; then, GH and HG indices are equal to
lattice of the TUC4C8(R)[a, b] nanosheet, where a and b are
parameters, is shown in Figure 8. Figure 9 shows the type 2 669 669GH􏼐T [a, b]􏼑 � 54ab + a + b + 16,
2 − C C (R) nanosheet. In the following theorem, GH, HG, 20 204 8
NGH, andNHG indices of this nanosheet are computed.)e (23)
number of vertices of the type-2 structure is 4 4 2 2 191 191ab + HG􏼐T [a, b]􏼑 � ab + a + b + 1.
(a + b) + 4, and the number of edges is 6ab + 5a + 5b + 4. 3 250 250
10 Journal of Chemistry
Figure 8: A TUC4C8(R)[a, b] nanotube.
1 2 3 a
1
2
3
b
Figure 9: Type II-C4C8(R) nanosheet T2[a, b].
Table 8: Edge partition of T2[a, b].
(du, dv) with uv ∈ E(G) Number of edges
(2, 2) 4
(2, 3) 4(a + b)
(3, 3) 6ab + a + b
Proof. Using Table 8, the definitions of GH and HG indices
are as follows:
􏽰������
2 ( du + dv􏼁 du × dGH T [a, b] � 􏽘 v􏼐 􏼑
uv∈ 2E(G)
√���� √���� √����
(2 + 2)( 2 × 2 ) (2 + 3)( 2 × 3 ) (3 + 3)( 3 × 3 )
� 4􏼨 􏼩 +(4a + 4b)􏼨 􏼩 +(6ab + a + b)􏼨 􏼩
2 2 2
2 669 669GH􏼐T [a, b]􏼑 � 54ab + a + b + 16,
20 20
(24)
2
HG 2􏼐T [a, b]􏼑 � 􏽘 􏽰������
( d
uv∈E(G) u + dv􏼁 du × dv
2 2 2
� 2􏼨 √���� 􏼩 +(4a + 4b)􏼨 √���� 􏼩 +(6ab + a + b)􏼨 √���� 􏼩
(2 + 2)( 2 × 2 ) (2 + 3)( 2 × 3 ) (3 + 3)( 3 × 3 )
2 2 191 191HG􏼐T [a, b]􏼑 � ab + a + b + 1.
3 250 250 □
Journal of Chemistry 11
)e edge partition of the T2[a, b] nanosheet based on the Table 9: Edge partition of T2[a, b].
neighborhood degree of vertices is detailed in Table 9.
(Su, Sv) with uv ∈ E(G) Number of edges
(5, 5) 4
Theorem 4. Let T2[a, b] be an (a, b)-dimensional nano- (5, 8) 8
sheet; then, NGH and NHG indices are equal to (6, 8) 4a + 4b − 8
2 20549 20549 21549
(8, 8) 2a + 2b + 4
NGH􏼐T [a, b]􏼑 � 486ab + a + b + , (8, 9) 4a + 4b − 8
100 100 500 (9, 9) 6ab − 5a − 5b + 4
2 37 107 107 19NHG􏼐T [a, b]􏼑 � ab + a + b + .
500 1000 1000 100
(25) Proof. Using Table 9, the definitions of NGH and NHG
indices are as follows:
􏽰������
2 ( Su + Sv􏼁 Su × SNGH v􏼐T [a, b]􏼑 � 􏽘
uv∈ 2E(G)
√���� √���� √����
(5 + 5)( 5 × 5 ) (5 + 8)( 5 × 8 ) (6 + 8)( 6 × 8 )
� 4􏼨 􏼩 + 8􏼨 􏼩 +(4a + 4b − 8)􏼨 􏼩 +(2a + 2b + 4)
2 2 2
√���� √���� √����
(8 + 8)( 8 × 8 ) (8 + 9)( 8 × 9 ) (9 + 9)( 9 × 9 )
· 􏼨 􏼩 +(4a + 4b − 8)􏼨 􏼩 +(6ab − 5a − 5b + 4)􏼨 􏼩
2 2 2
2 20549 20549 21549NGH􏼐T [a, b]􏼑􏼑 � 486ab + a + b + ,
100 100 500
NHG 2
2
􏼐T [a, b]􏼑 � 􏽘 􏽰������
∈ ( Su + S 􏼁 S × Suv E(G) v u v
2 2 2
� 4􏼨 √���� 􏼩 + 8􏼨 √���� 􏼩 +(4a + 4b − 8)􏼨 √���� 􏼩 +(2a + 2b + 4)
(5 + 5)( 5 × 5 ) (5 + 8)( 5 × 8 ) (6 + 8)( 6 × 8 )
2 2 2
· 􏼨 √���� 􏼩 +(4a + 4b − 8)􏼨 √���� 􏼩 +(6ab − 5a − 5b + 4)􏼨 √���� 􏼩
(8 + 8)( 8 × 8 ) (8 + 9)( 8 × 9 ) (9 + 9)( 9 × 9 )
2 37 107 107 19NHG􏼐T [a, b]􏼑 � ab + a + b + .
500 1000 1000 100
(26)
□
2.3.Results for theH-NaphthalenicNanosheet. Carbon atoms )e edge partition of the T3[a, b] nanosheet based on the
bonded in the form of a hexagonal structure constitute degree of vertices is detailed in Table 10.
carbon nanotubes. )ey are peri-condensed benzenoids
which mean three or more rings share the same atoms. 3
H-Naphthalenic nanosheet is constituted by the alternating Theorem 5. Let T [a, b] be an (a, b)-dimensional nano-
sequence of squares C , hexagons C , and octagons C and is sheet; then, GH and HG indices are equal to4 6 8
represented as T3[a, b], where a and b are the parameters. 3 4101 7901 301
)e number of vertices of the H-naphthalenic nanosheet is GH􏼐T [a, b]􏼑 � 135ab − a − b + ,100 200 100
10ab, and the edges are 15ab − 2a − 2b. )e GH, HG, NGH, (27)
and NHG indices of this nanosheet are computed; see 1667 39 33 69HG 3􏼐T [a, b]􏼑 � ab + a + b + .
Figure 10. 1000 200 125 1500
12 Journal of Chemistry
1 2 3 a
1
2
3
b
Figure 10: An H-naphthalenic nanosheet T3[a, b].
Table 10: )e details of edges and types of the T3[a, b] nanosheet based on the degree of vertices.
(du, dv) with uv ∈ E(G) Number of edges
(2, 2) 2a + 4
(2, 3) 8a + 4b − 8
(3, 3) 15ab − 10a − 8b + 4
Proof. Using Table 10, the definitions of GH and HG indices
are as follows:
􏽰������
GH 3
( d + d 􏼁 d × d
􏼐T [a, b] � 􏽘
u v u v
􏼑
∈ 2uv E(G)
√���� √����
(2 + 2)( 2 × 2 ) (2 + 3)( 2 × 3 )
� (2b + 4)􏼨 􏼩 +(8a + 8b − 8)􏼨 􏼩
2 2
√����
(3 + 3)( 3 × 3 )
+(15ab − 10a − 8b + 4)􏼨 􏼩
2
3 4101 7901 301GH􏼐T [a, b]􏼑 � 135ab − a − b + .
100 200 100
(28)
2
HG 3􏼐T [a, b]􏼑 � 􏽘 􏽰������
( d + d
uv∈E(G) u v􏼁 du × dv
2 2
� (2b + 4)􏼨 √���� 􏼩 +(8a + 8b − 8)􏼨 √���� 􏼩
(2 + 2)( 2 × 2 ) (2 + 3)( 2 × 3 )
2
+(15ab − 10a − 8b + 4)􏼨 √���� 􏼩
(3 + 3)( 3 × 3 )
HG 3
1667 39 33 69
􏼐T [a, b]􏼑 � ab + a + b + .
1000 200 125 1500 □
Journal of Chemistry 13
Table 11: Edge partition of T3[a, b].
(Su, Sv) with uv ∈ E(G) Number of edges
(4, 5) 8
(5, 5) 2b − 4
(5, 7) 4
(5, 8) 4b − 4
(6, 7) 4a − 4
(6, 8) 4a − 4
(7, 9) 2a
(8, 8) 2a + 2b − 4
(8, 9) 4a + 4b − 8
(9, 9) 15ab − 18a − 14b + 16
)e edge partition of the T3[a, b] nanosheet based on the Proof. Using Table 11, the definitions of NGH and NHG
neighborhood degree of vertices is detailed in Table 11. indices are as follows:
Theorem 6. Let T3[a, b] be an (a, b)-dimensional nano-
sheet; then, NGH and NHG indices are equal to
NGH 3
11199 251531 17382
􏼐T [a, b]􏼑 � 1215ab − a − b + ,
20 500 125
NHG 3
37 15 91 249
􏼐T [a, b]􏼑 � ab + a + b + .
200 200 1000 2500
(29)
􏽰������
( S + S 􏼁 S × S
NGH 3 u v u v􏼐T [a, b]􏼑 � 􏽘
2
uv∈E(G)
√���� √���� √����
(4 + 5)( 4 × 5 ) (5 + 5)( 5 × 5 ) (5 + 7)( 7 × 7 )
� 8􏼨 􏼩 +(2b − 4)􏼨 􏼩 + 4􏼨 􏼩 +(4b − 4)
2 2 2
√���� √���� √����
(5 + 8)( 5 × 8 ) (6 + 7)( 6 × 7 ) (6 + 8)( 6 × 8 )
· 􏼨 􏼩 +(4a − 4)􏼨 􏼩 +(4a − 4)􏼨 􏼩
2 2 2
√���� √����
(7 + 9)( 7 × 9 ) (8 + 8)( 8 × 8 )
+ 2a􏼨 􏼩 +(2a + 2b − 4)􏼨 􏼩 +(4a + 4b − 8)
2 2
√���� √����
(8 + 9)( 8 × 9 ) (9 + 9)( 9 × 9 )
· 􏼨 􏼩 +(15ab − 18a − 14b + 16)􏼨 􏼩
2 2
NGH 3
11199 251531 17382
􏼐T [a, b]􏼑 � 1215ab − a − b + ,
20 500 125
􏽰������ (30)
3 ( Su + Sv􏼁 Su × SNHG v􏼐T [a, b]􏼑 � 􏽘
2
uv∈E(G)
2 2 2
� 8􏼨 √���� 􏼩 +(2b − 4)􏼨 √���� 􏼩 + 4􏼨 √���� 􏼩 +(4b − 4)
(4 + 5)( 4 × 5 ) (5 + 5)( 5 × 5 ) (5 + 7)( 7 × 7 )
2 2 2
· 􏼨 √���� 􏼩 +(4a − 4)􏼨 √���� 􏼩 +(4a − 4)􏼨 √���� 􏼩
(5 + 8)( 5 × 8 ) (6 + 7)( 6 × 7 ) (6 + 8)( 6 × 8 )
2 2
+ 2a􏼨 √���� 􏼩 +(2a + 2b − 4)􏼨 √���� 􏼩 +(4a + 4b − 8)
(7 + 9)( 7 × 9 ) (8 + 8)( 8 × 8 )
2 2
· 􏼨 √���� 􏼩 +(15ab − 18a − 14b + 16)􏼨 √���� 􏼩
(8 + 9)( 8 × 9 ) (9 + 9)( 9 × 9 )
37 15 91 249
NHG 3􏼐T [a, b]􏼑 � ab + a + b + .
200 200 1000 2500 □
14 Journal of Chemistry
3. Conclusion of Applied Mathematics and Computing, vol. 66, pp. 1–14,
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