# Novel Degree-Based Topological Descriptors of Carbon Nanotubes

Authors: M. C. Shanmukha; A. Usha; M. K. Siddiqui; K. C. Shilpa; A. Asare-Tuah
Journal: Journal of Chemistry
 (2021)

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## Abstract

The 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.


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## Body

## 1. Introduction

The applications of graph theory are diversified in every field, but chemistry is the major area of the implementation of graph theory. In chemical graph theory, topological index plays a vital role which facilitates the chemists with a treasure of data that correlate with the structure of the chemical compound. The topological index is a numerical descriptor, defines the graph topology of the molecule, and predicts an extensive range of molecular properties5[1–6].From the last two decades, topological indices (TIs) are identified and used in pharmacological medicine, bioinorganic chemistry, toxicity, and theoretical chemistry and are also used for correlation analysis [7–11].Topological descriptors are frequently used in the discovery of drugs as they have rich datasets that give high predictive values. These descriptors give the information depending on the arrangement of atoms and their bonds of a chemical compound. They are studied for chemical compounds where, generally, the hydrogen atoms are suppressed. The originality of QSAR/QSPR models depends on physicochemical properties for chemical compounds with high degree of precision. These models depend on various factors such as selecting the suitable compounds, suitable descriptors, and suitable algorithms or tools used in model development [12]. The QSAR/QSPR analysis is based on the data obtained by the numerical descriptors. These data are used to verify whether the compound under the study is suitable for drug making as the TIs provide computational data about the compound. Considering the information of the compound, QSAR/QSPR/QSTR analyses are carried out.The TIs have increasing popularity in the field of research as they involve only computation without performing any physical experiment. Recent years have proved considerable attention in TIs as the effects of an atomic type and group efforts are considered in QSAR/QSPR modelling [13–15]. Distance-based TIs are used in QSAR analysis, while chirality descriptors are introduced based on molecular graphs [16].Alkanes are acyclic saturated hydrocarbons in which carbons and hydrogens are arranged in a tree-like structure. The main use of alkanes is found in crude oil such as petroleum, cooking gas, pesticides, and drug synthesis. The compounds that contain absolutely the same number of atoms but their arrangement differs are termed as isomers. A study is carried out for eighteen octane isomers (refer Figure1).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-methylpentane, (m) 3-ethyl-3-methylpentane, (n) 2,2,3-trimethylpentane, (o) 2,2,4-trimethylpentane, (p) 2,3,3-trimethylpentane, (q) 2,3,4-trimethylpentane, and (r) 2,2,3,3-trimethylbutane.
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)(m)(n)(o)(p)(q)(r)A structure whose size is between the microscopic and molecular structure is referred to as a nanostructure. There are different types of nanostructures, namely, nanocages, nanocomposites, nanoparticles, nanofabrics, etc. In the recent years, nanostructures have attracted a lot of researchers in the areas of biology, chemistry, and medicines. Topological indices of nanostructures can be studied from [17–24]. The nanostructures made of carbons with cylindrical shape are carbon nanotubes (CNTs). They have a similar structure to that of a fullerene and graphene except their cylindrical shape. The shape of fullerene is as that of a football or basketball design where hexagons are connected.In 1991, Iijima [25] used carbon nanotubes that have attracted many researchers in nanoscience and nanotechnology worldwide. As they have exotic properties, they are widely used in both research and applications. Nanotubes have a distinctive structure with remarkable mechanical and electrical properties. In case of carbon nanotubes, the hexagons are surrounded by squares, and each of these patterns is linearly arranged. Carbon nanotubes reveal exceptional electrical conductivity and possess wonderful tensile strength and thermal conductivity as they have nanostructures in which the carbon atoms are strongly connected.Carbon nanotubes have applications in orthopaedic implants, especially in total hip replacement and other treatments pertaining to bone-related ailments. They are used as a grouting agent placed between the prosthesis and the bone as a part of their therapeutic use. The CNTs are used in biomedical fields because of their structural stiffness and effective optical absorption from UV to IR. Also, they can be altered chemically which are expected to be useful in many fields of technology such as electronics, composited materials, and carbon fibres. They have incredible applications in 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 along its own axis. Even though nanotubes have favourable applications in a variety of fields, their large-scale production has been restricted. The main constraint that obstructs their use lies in difficulty in controlling their structure, impurities, and poor process ability. To enhance their usage, they have grabbed the attention especially in the formation of composites with polymers.There are two types of configurations in the arrangement of nanotubes, namely, zigzag and armchair. In the zigzag configuration, the hexagons are placed one below the other linearly, whereas in the armchair configuration, they are placed next to each other. This gives two different types of configurations with different terminologies discussed now. To explain the structure of a nanotube that is infinitely long, we imagine it to be cut open by a parallel axis and placed on a plane. Then, the atoms and bonds coincide with an imaginary graphene sheet. The length of the two atoms on opposite edges of the strip corresponds to circumference of the cylindrical graphene sheet [27–29].The main objectives of this work are as follows:To define novel indicesTo discuss the physical and chemical applicability of octane isomers using regression modelsTo compute defined indices for carbon nanotubes such asC4C8S, C4C8R, and H-naphthalenic nanosheetsLetG=V,E be a graph with a vertex set VG and an edge set EG such that VG=n and EG=m. For standard graph terminologies and notations, refer to [30, 31].where u,v is an element of EG, du represents the degree of the vertex u, and Su represents the neighborhood degree of the vertex u.Definition 1.
Recently, Usha et al. [32] defined the geometric-harmonic (GH) index, inspired by Vukicevic and Furtula [33] in designing the GA index:(1)GHG=∑du+dvdu⋅dv2.
Motivated by the above work, in this paper, an attempt is made to define three novel indices based on degree and neighborhood degree, namely, harmonic-geometric (HG), neighborhood geometric-harmonic (NGH), and neighborhood harmonic-geometric (NHG) indices. They are defined as follows:(2)HGG=∑2du+dvdu⋅dv,NGHG=∑Su+SvSu⋅Sv2,NHGG=∑2Su+SvSu⋅Sv,


### 1.1. Chemical Applicability ofGH, NGH, HG, and NHG Indices

In this section, a linear regression model of four physical properties is presented forGH, NGH, HG, and NHG indices. The physical properties such as entropyS, acentric factor AF, enthalpy of vaporization HVAP, and standard enthalpy of vaporization DHVAP of octane isomers have shown good correlation with the indices considered in the study. The GH, HG, NGH, and NHG indices are tested for the octane isomers’ database available at https://www.moleculardescriptors.eu/dataset.htm. The GH, HG, NGH, and NHG indices are computed and tabulated in columns 6, 7, 8, and 9 of Table 1.Table 1
Experimental values ofS, AF, HVAP, and DHVAP and the corresponding values of the GH index, HG index, NGH index, and NHG index of octane isomers.
AlkaneSAFHVAPDHVAPGHNGHHGNHGn-Octane111.7000.39873.1909.91524.42384.4962.1930.6792-Methylheptane109.8000.37870.3009.48427.17398.7461.9620.5733-Methylheptane111.3000.37171.3009.52127.954107.4752.0580.5684-Methylheptane109.3000.37270.9109.48327.954113.3562.0580.6003-Ethylhexane109.4000.36271.7009.47628.735131.9412.1540.4762,2-Dimethylhexane103.4000.33967.7008.91533.607122.9701.6890.4712,3-Dimethylhexane108.0000.34870.2009.27231.637110.6221.8620.4982,4-Dimethylhexane107.0000.34468.5009.02930.885123.2361.8270.4742,5-Dimethylhexane105.7000.35768.6009.05132.248133.2421.7310.4693,3-Dimethylhexane104.7000.32368.5008.97335.213151.3981.8290.4493,4-Dimethylhexane106.6000.34070.2009.31632.418117.7011.9580.5782-Methyl-3-ethylpentane106.1000.33269.7009.20932.418176.1111.9580.3423-Methyl-3-ethylpentane101.5000.30769.3009.08136.820149.2221.9680.3772,2,3-Trimethylpentane101.3000.30167.3008.82638.834164.3661.6060.3382,2,4-Trimethylpentane104.1000.30564.8708.40236.537155.8741.4590.3582,3,3-Trimethylpentane102.1000.29368.1008.89739.659141.6571.6490.4622,3,4-Trimethylpentane102.4000.31768.3709.01435.321168.7971.6660.3902,2,3,3-Trimethylbutane93.0600.25566.2008.41046.000223.6201.2630.227Using the method of least squares, the linear regression models for S, AF, HVAP, and DHVAP are fitted using the data of Table1.The fitted models for theGH index are(3)S=133.078±1.82−0.833±0.054GH,(4)acentric factor=0.557±0.009−0.007±0.000GH,(5)HVAP=79.613±1.878−0.315±0.056GH,(6)DHVAP=11.273±0.285−0.065±0.008GH.The fitted models for theHG index are(7)S=76.608±4.486+15.766±2.435HG,(8)acentric factor=0.114±0.037+0.121±0.020HG,(9)HVAP=54.960±1.356+7.773±0.736HG,(10)DHVAP=6.428±0.249+1.477±0.135HG.The fitted models for theNGH index are(11)S=121.77±2.35−0.119±0.017NGH,(12)acentric factor=0.465±0.018−0.001±0.000NGH,(13)HVAP=75.007±1.552−0.043±0.011NGH,(14)DHVAP=10.363±0.257−0.009±0.002NGH.The fitted models for theNHG index are(15)S=89.524±2.505+34.35±5.271NHG,(16)acentric factor=0.207±0.018+0.277±0.038NHG,(17)HVAP=62.667±1.352+14.044±2.846NHG,(18)DHVAP=7.793±0.219+2.882±0.460NHG.Note: in equations (3)–(18), the errors of the regression coefficients are represented within brackets.Tables2–5 and Figures 2–5 show the correlation coefficient and residual standard error for the regression models of four physical properties with GH, HG, NGH, and NHG indices.Table 2
Parameters of regression models for theGH index.
Physical propertiesValue of the correlation coefficientResidual standard errorEntropy−0.9681.17Acentric factor−0.9870.0059HVAP−0.8151.21DHVAP−0.8850.184Table 3
Parameters of regression models for theHG index.
Physical propertiesValue of the correlation coefficientResidual standard errorEntropy0.852.448Acentric factor0.8330.020HVAP0.9350.74DHVAP0.9390.136Table 4
Parameters of regression models for theNGH index.
Physical propertiesValue of the correlation coefficientResidual standard errorEntropy−0.8732.274Acentric factor−0.8770.0176HVAP−0.6951.502DHVAP−0.7780.248Table 5
Parameters of regression models for theNHG index.
Physical propertiesValue of the correlation coefficientResidual standard errorEntropy0.8522.436Acentric factor0.8770.018HVAP0.7771.315DHVAP0.8430.213Figure 2
Scatter diagram of physical propertiesS, AF, HVAP, and DHVAP with the GH index.
(a)(b)(c)(d)Figure 3
Scatter diagram of physical propertiesS, AF, HVAP, and DHVAP with the HG index.
(a)(b)(c)(d)Figure 4
Scatter diagram of physical propertiesS, AF, HVAP, and DHVAP with the NGH index.
(a)(b)(c)(d)Figure 5
Scatter diagram of physical propertiesS, AF, HVAP, and DHVAP with the NHG index.
(a)(b)(c)(d)From Table2 and Figure 2, it is obvious that the GH index highly correlates with the acentric factor and the correlation coefficient r=0.987. Also, the GH index has good correlation coefficient r=0.968 with entropy, r=0.815 with HVAP, and r=0.885 with DHVAP.From Table3 and Figure 3, it is noticed that the HG index highly correlates with DHVAP and the correlation coefficient r=0.939. Also, the HG index has good correlation coefficient r=0.85 with entropy, r=0.833 with the acentric factor, and r=0.935 with HVAP.From Table4 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 correlation coefficient r=0.873 with entropy, r=0.695 with HVAP, and r=0.778 with DHVAP.From Table5 and Figure 5, it is clear that the NHG index highly correlates with the acentric factor and the correlation coefficient r=0.877. Also, the NHG index has good correlation coefficient r=0.852 with entropy, r=0.777 with HVAP, and r=0.843 with DHVAP.


## 1.1. Chemical Applicability ofGH, NGH, HG, and NHG Indices

In this section, a linear regression model of four physical properties is presented forGH, NGH, HG, and NHG indices. The physical properties such as entropyS, acentric factor AF, enthalpy of vaporization HVAP, and standard enthalpy of vaporization DHVAP of octane isomers have shown good correlation with the indices considered in the study. The GH, HG, NGH, and NHG indices are tested for the octane isomers’ database available at https://www.moleculardescriptors.eu/dataset.htm. The GH, HG, NGH, and NHG indices are computed and tabulated in columns 6, 7, 8, and 9 of Table 1.Table 1
Experimental values ofS, AF, HVAP, and DHVAP and the corresponding values of the GH index, HG index, NGH index, and NHG index of octane isomers.
AlkaneSAFHVAPDHVAPGHNGHHGNHGn-Octane111.7000.39873.1909.91524.42384.4962.1930.6792-Methylheptane109.8000.37870.3009.48427.17398.7461.9620.5733-Methylheptane111.3000.37171.3009.52127.954107.4752.0580.5684-Methylheptane109.3000.37270.9109.48327.954113.3562.0580.6003-Ethylhexane109.4000.36271.7009.47628.735131.9412.1540.4762,2-Dimethylhexane103.4000.33967.7008.91533.607122.9701.6890.4712,3-Dimethylhexane108.0000.34870.2009.27231.637110.6221.8620.4982,4-Dimethylhexane107.0000.34468.5009.02930.885123.2361.8270.4742,5-Dimethylhexane105.7000.35768.6009.05132.248133.2421.7310.4693,3-Dimethylhexane104.7000.32368.5008.97335.213151.3981.8290.4493,4-Dimethylhexane106.6000.34070.2009.31632.418117.7011.9580.5782-Methyl-3-ethylpentane106.1000.33269.7009.20932.418176.1111.9580.3423-Methyl-3-ethylpentane101.5000.30769.3009.08136.820149.2221.9680.3772,2,3-Trimethylpentane101.3000.30167.3008.82638.834164.3661.6060.3382,2,4-Trimethylpentane104.1000.30564.8708.40236.537155.8741.4590.3582,3,3-Trimethylpentane102.1000.29368.1008.89739.659141.6571.6490.4622,3,4-Trimethylpentane102.4000.31768.3709.01435.321168.7971.6660.3902,2,3,3-Trimethylbutane93.0600.25566.2008.41046.000223.6201.2630.227Using the method of least squares, the linear regression models for S, AF, HVAP, and DHVAP are fitted using the data of Table1.The fitted models for theGH index are(3)S=133.078±1.82−0.833±0.054GH,(4)acentric factor=0.557±0.009−0.007±0.000GH,(5)HVAP=79.613±1.878−0.315±0.056GH,(6)DHVAP=11.273±0.285−0.065±0.008GH.The fitted models for theHG index are(7)S=76.608±4.486+15.766±2.435HG,(8)acentric factor=0.114±0.037+0.121±0.020HG,(9)HVAP=54.960±1.356+7.773±0.736HG,(10)DHVAP=6.428±0.249+1.477±0.135HG.The fitted models for theNGH index are(11)S=121.77±2.35−0.119±0.017NGH,(12)acentric factor=0.465±0.018−0.001±0.000NGH,(13)HVAP=75.007±1.552−0.043±0.011NGH,(14)DHVAP=10.363±0.257−0.009±0.002NGH.The fitted models for theNHG index are(15)S=89.524±2.505+34.35±5.271NHG,(16)acentric factor=0.207±0.018+0.277±0.038NHG,(17)HVAP=62.667±1.352+14.044±2.846NHG,(18)DHVAP=7.793±0.219+2.882±0.460NHG.Note: in equations (3)–(18), the errors of the regression coefficients are represented within brackets.Tables2–5 and Figures 2–5 show the correlation coefficient and residual standard error for the regression models of four physical properties with GH, HG, NGH, and NHG indices.Table 2
Parameters of regression models for theGH index.
Physical propertiesValue of the correlation coefficientResidual standard errorEntropy−0.9681.17Acentric factor−0.9870.0059HVAP−0.8151.21DHVAP−0.8850.184Table 3
Parameters of regression models for theHG index.
Physical propertiesValue of the correlation coefficientResidual standard errorEntropy0.852.448Acentric factor0.8330.020HVAP0.9350.74DHVAP0.9390.136Table 4
Parameters of regression models for theNGH index.
Physical propertiesValue of the correlation coefficientResidual standard errorEntropy−0.8732.274Acentric factor−0.8770.0176HVAP−0.6951.502DHVAP−0.7780.248Table 5
Parameters of regression models for theNHG index.
Physical propertiesValue of the correlation coefficientResidual standard errorEntropy0.8522.436Acentric factor0.8770.018HVAP0.7771.315DHVAP0.8430.213Figure 2
Scatter diagram of physical propertiesS, AF, HVAP, and DHVAP with the GH index.
(a)(b)(c)(d)Figure 3
Scatter diagram of physical propertiesS, AF, HVAP, and DHVAP with the HG index.
(a)(b)(c)(d)Figure 4
Scatter diagram of physical propertiesS, AF, HVAP, and DHVAP with the NGH index.
(a)(b)(c)(d)Figure 5
Scatter diagram of physical propertiesS, AF, HVAP, and DHVAP with the NHG index.
(a)(b)(c)(d)From Table2 and Figure 2, it is obvious that the GH index highly correlates with the acentric factor and the correlation coefficient r=0.987. Also, the GH index has good correlation coefficient r=0.968 with entropy, r=0.815 with HVAP, and r=0.885 with DHVAP.From Table3 and Figure 3, it is noticed that the HG index highly correlates with DHVAP and the correlation coefficient r=0.939. Also, the HG index has good correlation coefficient r=0.85 with entropy, r=0.833 with the acentric factor, and r=0.935 with HVAP.From Table4 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 correlation coefficient r=0.873 with entropy, r=0.695 with HVAP, and r=0.778 with DHVAP.From Table5 and Figure 5, it is clear that the NHG index highly correlates with the acentric factor and the correlation coefficient r=0.877. Also, the NHG index has good correlation coefficient r=0.852 with entropy, r=0.777 with HVAP, and r=0.843 with DHVAP.


## 2. GH, NGH, HG, and NHG Indices ofC4C8S, C4C8R, and H-Naphthalenic Nanosheets

### 2.1. Results for theC4C8S Nanosheet

The alternating pattern of 4 carbon atoms forming squares and 8 carbon atoms forming octagons constitutes theTUC4C8Sa,b nanosheet.In this section,GH, HG, NGH, and NHG indices of the C4C8S nanosheet are computed. The pattern of carbon atoms gives rise to two types of nanosheets, namely, T1a,b and T2a,b. The 2-dimensional nanosheet is represented by T1a,b, where a and b are parameters (Figure 6). In T1a,b, C4 acts as a square, while C8 is an octagon in which a and b represent the column and row, respectively. Figure 7 depicts the type 1−C4C8S nanosheet. The number of vertices of the C4C8S nanosheet is 8ab, and the number of edges is 12ab−2a−2b.Figure 6
ATUC4C8Sa,b nanotube.Figure 7
Type I-C4C8S nanosheet T1a,b.The edge partition of theT1a,b nanosheet based on the degree of vertices is detailed in Table 6.Table 6
The edge partition ofT1a,b.
du,dv with uv∈EGNumber of edges2,22a+b+22,34a+4b−83,312ab−8a−8b+4Theorem 1.
LetT1a,b be an a,b-dimensional nanosheet; then, GH and HG indices are equal to(19)GHT1a,b=108ab−4938125a−4938125b+376125,HGT1a,b=43ab+33125a+33125b+69500.Proof.
Using Table6, the definitions of GH and HG indices are as follows:(20)GHT1a,b=∑uv∈EGdu+dvdu×dv2=2a+2b+42+22×22+4a+4b−82+32×32+12ab−8a−8b+42+32×32GHT1a,b=108ab−4938125a−4938125b+376125,HGT1a,b=∑uv∈EG2du+dvdu×dv=2a+2b+422+22×2+4a+4b−822+32×3+12ab−8a−8b+422+32×3HGT1a,b=43ab+33125a+33125b+69500.The edge partition of theT1a,b nanosheet based on the neighborhood degree of vertices is detailed in Table 7.Table 7
Edge partition ofT1a,b for neighborhood degree-based vertices.
Su,Sv with uv∈EGNumber of edges4,444,585,52a+2b−85,84a+4b−88,82a+2b−48,94a+4b−89,912ab−14a−14b+16Theorem 2.
LetT1a,b be an a,b-dimensional nanosheet; then, NGH and NHG indices are equal to(21)NGHT1a,b=972ab−251531500a−251531500b+1591211000,NHGT1a,b=37250ab+911000a+911000b+1571000.Proof.
Using Table7, the definitions of NGH and NHG indices are as follows:(22)NGHT1a,b=∑uv∈EGSu+SvSu×Sv2=44+44×42+84+54×52+2a+2b−85+55×52+4a+4b−85+85×82+2a+2b−48+88×82+4a+4b−88+98×92+12ab−14a−14b+169+99×92NGHT1a,b=972ab−251531500a−251531500b+1591211000,NHGT1a,b=∑uv∈EGSu+SvSu×Sv2=424+44×4+824+54×5+2a+2b−825+55×5+4a+4b−825+85×8+2a+2b−428+88×8+4a+4b−828+98×9+12ab−14a−14b+1629+99×9NHGT1a,b=37250ab+911000a+911000b+1571000.


### 2.2. Results for theC4C8R Nanosheet

This structure is formed by 4 carbon atoms forming a rhombus that are 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 octagons and is represented asT2a,b. The 2-dimensional lattice of the TUC4C8Ra,b nanosheet, where a and b are parameters, is shown in Figure 8. Figure 9 shows the type 2−C4C8R nanosheet. In the following theorem, GH, HG, NGH, and NHG indices of this nanosheet are computed. The number of vertices of the type-2 structure is 4ab+4a+b+4, and the number of edges is 6ab+5a+5b+4.Figure 8
ATUC4C8Ra,b nanotube.Figure 9
Type II-C4C8R nanosheet T2a,b.The edge partition of theT2a,b nanosheet for degree-based vertices is detailed in Table 8.Table 8
Edge partition ofT2a,b.
du,dv with uv∈EGNumber of edges2,242,34a+b3,36ab+a+bTheorem 3.
LetT2a,b be an a,b-dimensional nanosheet; then, GH and HG indices are equal to(23)GHT2a,b=54ab+66920a+66920b+16,HGT2a,b=23ab+191250a+191250b+1.Proof.
Using Table8, the definitions of GH and HG indices are as follows:(24)GHT2a,b=∑uv∈EGdu+dvdu×dv2=42+22×22+4a+4b2+32×32+6ab+a+b3+33×32GHT2a,b=54ab+66920a+66920b+16,HGT2a,b=∑uv∈EG2du+dvdu×dv=222+22×2+4a+4b22+32×3+6ab+a+b23+33×3HGT2a,b=23ab+191250a+191250b+1.The edge partition of theT2a,b nanosheet based on the neighborhood degree of vertices is detailed in Table 9.Table 9
Edge partition ofT2a,b.
Su,Sv with uv∈EGNumber of edges5,545,886,84a+4b−88,82a+2b+48,94a+4b−89,96ab−5a−5b+4Theorem 4.
LetT2a,b be an a,b-dimensional nanosheet; then, NGH and NHG indices are equal to(25)NGHT2a,b=486ab+20549100a+20549100b+21549500,NHGT2a,b=37500ab+1071000a+1071000b+19100.Proof.
Using Table9, the definitions of NGH and NHG indices are as follows:(26)NGHT2a,b=∑uv∈EGSu+SvSu×Sv2=45+55×52+85+85×82+4a+4b−86+86×82+2a+2b+48+88×82+4a+4b−88+98×92+6ab−5a−5b+49+99×92NGHT2a,b=486ab+20549100a+20549100b+21549500,NHGT2a,b=∑uv∈EG2Su+SvSu×Sv=425+55×5+825+85×8+4a+4b−826+86×8+2a+2b+428+88×8+4a+4b−828+98×9+6ab−5a−5b+429+99×9NHGT2a,b=37500ab+1071000a+1071000b+19100.


### 2.3. Results for the H-Naphthalenic Nanosheet

Carbon atoms bonded in the form of a hexagonal structure constitute carbon nanotubes. They are peri-condensed benzenoids which mean three or more rings share the same atoms. H-Naphthalenic nanosheet is constituted by the alternating sequence of squaresC4, hexagons C6, and octagons C8 and is represented as T3a,b, where a and b are the parameters. The number of vertices of the H-naphthalenic nanosheet is 10ab, and the edges are 15ab−2a−2b. The GH, HG, NGH, and NHG indices of this nanosheet are computed; see Figure 10.Figure 10
An H-naphthalenic nanosheetT3a,b.The edge partition of theT3a,b nanosheet based on the degree of vertices is detailed in Table 10.Table 10
The details of edges and types of theT3a,b nanosheet based on the degree of vertices.
du,dv with uv∈EGNumber of edges2,22a+42,38a+4b−83,315ab−10a−8b+4Theorem 5.
LetT3a,b be an a,b-dimensional nanosheet; then, GH and HG indices are equal to(27)GHT3a,b=135ab−4101100a−7901200b+301100,HGT3a,b=16671000ab+39200a+33125b+691500.Proof.
Using Table10, the definitions of GH and HG indices are as follows:(28)GHT3a,b=∑uv∈EGdu+dvdu×dv2=2b+42+22×22+8a+8b−82+32×32+15ab−10a−8b+43+33×32GHT3a,b=135ab−4101100a−7901200b+301100.HGT3a,b=∑uv∈EG2du+dvdu×dv=2b+422+22×2+8a+8b−822+32×3+15ab−10a−8b+423+33×3HGT3a,b=16671000ab+39200a+33125b+691500.The edge partition of theT3a,b nanosheet based on the neighborhood degree of vertices is detailed in Table 11.Table 11
Edge partition ofT3a,b.
Su,Sv with uv∈EGNumber of edges4,585,52b−45,745,84b−46,74a−46,84a−47,92a8,82a+2b−48,94a+4b−89,915ab−18a−14b+16Theorem 6.
LetT3a,b be an a,b-dimensional nanosheet; then, NGH and NHG indices are equal to(29)NGHT3a,b=1215ab−1119920a−251531500b+17382125,NHGT3a,b=37200ab+15200a+911000b+2492500.Proof.
Using Table11, the definitions of NGH and NHG indices are as follows:(30)NGHT3a,b=∑uv∈EGSu+SvSu×Sv2=84+54×52+2b−45+55×52+45+77×72+4b−45+85×82+4a−46+76×72+4a−46+86×82+2a7+97×92+2a+2b−48+88×82+4a+4b−88+98×92+15ab−18a−14b+169+99×92NGHT3a,b=1215ab−1119920a−251531500b+17382125,NHGT3a,b=∑uv∈EGSu+SvSu×Sv2=824+54×5+2b−425+55×5+425+77×7+4b−425+85×8+4a−426+76×7+4a−426+86×8+2a27+97×9+2a+2b−428+88×8+4a+4b−828+98×9+15ab−18a−14b+1629+99×9NHGT3a,b=37200ab+15200a+911000b+2492500.


## 2.1. Results for theC4C8S Nanosheet

The alternating pattern of 4 carbon atoms forming squares and 8 carbon atoms forming octagons constitutes theTUC4C8Sa,b nanosheet.In this section,GH, HG, NGH, and NHG indices of the C4C8S nanosheet are computed. The pattern of carbon atoms gives rise to two types of nanosheets, namely, T1a,b and T2a,b. The 2-dimensional nanosheet is represented by T1a,b, where a and b are parameters (Figure 6). In T1a,b, C4 acts as a square, while C8 is an octagon in which a and b represent the column and row, respectively. Figure 7 depicts the type 1−C4C8S nanosheet. The number of vertices of the C4C8S nanosheet is 8ab, and the number of edges is 12ab−2a−2b.Figure 6
ATUC4C8Sa,b nanotube.Figure 7
Type I-C4C8S nanosheet T1a,b.The edge partition of theT1a,b nanosheet based on the degree of vertices is detailed in Table 6.Table 6
The edge partition ofT1a,b.
du,dv with uv∈EGNumber of edges2,22a+b+22,34a+4b−83,312ab−8a−8b+4Theorem 1.
LetT1a,b be an a,b-dimensional nanosheet; then, GH and HG indices are equal to(19)GHT1a,b=108ab−4938125a−4938125b+376125,HGT1a,b=43ab+33125a+33125b+69500.Proof.
Using Table6, the definitions of GH and HG indices are as follows:(20)GHT1a,b=∑uv∈EGdu+dvdu×dv2=2a+2b+42+22×22+4a+4b−82+32×32+12ab−8a−8b+42+32×32GHT1a,b=108ab−4938125a−4938125b+376125,HGT1a,b=∑uv∈EG2du+dvdu×dv=2a+2b+422+22×2+4a+4b−822+32×3+12ab−8a−8b+422+32×3HGT1a,b=43ab+33125a+33125b+69500.The edge partition of theT1a,b nanosheet based on the neighborhood degree of vertices is detailed in Table 7.Table 7
Edge partition ofT1a,b for neighborhood degree-based vertices.
Su,Sv with uv∈EGNumber of edges4,444,585,52a+2b−85,84a+4b−88,82a+2b−48,94a+4b−89,912ab−14a−14b+16Theorem 2.
LetT1a,b be an a,b-dimensional nanosheet; then, NGH and NHG indices are equal to(21)NGHT1a,b=972ab−251531500a−251531500b+1591211000,NHGT1a,b=37250ab+911000a+911000b+1571000.Proof.
Using Table7, the definitions of NGH and NHG indices are as follows:(22)NGHT1a,b=∑uv∈EGSu+SvSu×Sv2=44+44×42+84+54×52+2a+2b−85+55×52+4a+4b−85+85×82+2a+2b−48+88×82+4a+4b−88+98×92+12ab−14a−14b+169+99×92NGHT1a,b=972ab−251531500a−251531500b+1591211000,NHGT1a,b=∑uv∈EGSu+SvSu×Sv2=424+44×4+824+54×5+2a+2b−825+55×5+4a+4b−825+85×8+2a+2b−428+88×8+4a+4b−828+98×9+12ab−14a−14b+1629+99×9NHGT1a,b=37250ab+911000a+911000b+1571000.


## 2.2. Results for theC4C8R Nanosheet

This structure is formed by 4 carbon atoms forming a rhombus that are 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 octagons and is represented asT2a,b. The 2-dimensional lattice of the TUC4C8Ra,b nanosheet, where a and b are parameters, is shown in Figure 8. Figure 9 shows the type 2−C4C8R nanosheet. In the following theorem, GH, HG, NGH, and NHG indices of this nanosheet are computed. The number of vertices of the type-2 structure is 4ab+4a+b+4, and the number of edges is 6ab+5a+5b+4.Figure 8
ATUC4C8Ra,b nanotube.Figure 9
Type II-C4C8R nanosheet T2a,b.The edge partition of theT2a,b nanosheet for degree-based vertices is detailed in Table 8.Table 8
Edge partition ofT2a,b.
du,dv with uv∈EGNumber of edges2,242,34a+b3,36ab+a+bTheorem 3.
LetT2a,b be an a,b-dimensional nanosheet; then, GH and HG indices are equal to(23)GHT2a,b=54ab+66920a+66920b+16,HGT2a,b=23ab+191250a+191250b+1.Proof.
Using Table8, the definitions of GH and HG indices are as follows:(24)GHT2a,b=∑uv∈EGdu+dvdu×dv2=42+22×22+4a+4b2+32×32+6ab+a+b3+33×32GHT2a,b=54ab+66920a+66920b+16,HGT2a,b=∑uv∈EG2du+dvdu×dv=222+22×2+4a+4b22+32×3+6ab+a+b23+33×3HGT2a,b=23ab+191250a+191250b+1.The edge partition of theT2a,b nanosheet based on the neighborhood degree of vertices is detailed in Table 9.Table 9
Edge partition ofT2a,b.
Su,Sv with uv∈EGNumber of edges5,545,886,84a+4b−88,82a+2b+48,94a+4b−89,96ab−5a−5b+4Theorem 4.
LetT2a,b be an a,b-dimensional nanosheet; then, NGH and NHG indices are equal to(25)NGHT2a,b=486ab+20549100a+20549100b+21549500,NHGT2a,b=37500ab+1071000a+1071000b+19100.Proof.
Using Table9, the definitions of NGH and NHG indices are as follows:(26)NGHT2a,b=∑uv∈EGSu+SvSu×Sv2=45+55×52+85+85×82+4a+4b−86+86×82+2a+2b+48+88×82+4a+4b−88+98×92+6ab−5a−5b+49+99×92NGHT2a,b=486ab+20549100a+20549100b+21549500,NHGT2a,b=∑uv∈EG2Su+SvSu×Sv=425+55×5+825+85×8+4a+4b−826+86×8+2a+2b+428+88×8+4a+4b−828+98×9+6ab−5a−5b+429+99×9NHGT2a,b=37500ab+1071000a+1071000b+19100.


## 2.3. Results for the H-Naphthalenic Nanosheet

Carbon atoms bonded in the form of a hexagonal structure constitute carbon nanotubes. They are peri-condensed benzenoids which mean three or more rings share the same atoms. H-Naphthalenic nanosheet is constituted by the alternating sequence of squaresC4, hexagons C6, and octagons C8 and is represented as T3a,b, where a and b are the parameters. The number of vertices of the H-naphthalenic nanosheet is 10ab, and the edges are 15ab−2a−2b. The GH, HG, NGH, and NHG indices of this nanosheet are computed; see Figure 10.Figure 10
An H-naphthalenic nanosheetT3a,b.The edge partition of theT3a,b nanosheet based on the degree of vertices is detailed in Table 10.Table 10
The details of edges and types of theT3a,b nanosheet based on the degree of vertices.
du,dv with uv∈EGNumber of edges2,22a+42,38a+4b−83,315ab−10a−8b+4Theorem 5.
LetT3a,b be an a,b-dimensional nanosheet; then, GH and HG indices are equal to(27)GHT3a,b=135ab−4101100a−7901200b+301100,HGT3a,b=16671000ab+39200a+33125b+691500.Proof.
Using Table10, the definitions of GH and HG indices are as follows:(28)GHT3a,b=∑uv∈EGdu+dvdu×dv2=2b+42+22×22+8a+8b−82+32×32+15ab−10a−8b+43+33×32GHT3a,b=135ab−4101100a−7901200b+301100.HGT3a,b=∑uv∈EG2du+dvdu×dv=2b+422+22×2+8a+8b−822+32×3+15ab−10a−8b+423+33×3HGT3a,b=16671000ab+39200a+33125b+691500.The edge partition of theT3a,b nanosheet based on the neighborhood degree of vertices is detailed in Table 11.Table 11
Edge partition ofT3a,b.
Su,Sv with uv∈EGNumber of edges4,585,52b−45,745,84b−46,74a−46,84a−47,92a8,82a+2b−48,94a+4b−89,915ab−18a−14b+16Theorem 6.
LetT3a,b be an a,b-dimensional nanosheet; then, NGH and NHG indices are equal to(29)NGHT3a,b=1215ab−1119920a−251531500b+17382125,NHGT3a,b=37200ab+15200a+911000b+2492500.Proof.
Using Table11, the definitions of NGH and NHG indices are as follows:(30)NGHT3a,b=∑uv∈EGSu+SvSu×Sv2=84+54×52+2b−45+55×52+45+77×72+4b−45+85×82+4a−46+76×72+4a−46+86×82+2a7+97×92+2a+2b−48+88×82+4a+4b−88+98×92+15ab−18a−14b+169+99×92NGHT3a,b=1215ab−1119920a−251531500b+17382125,NHGT3a,b=∑uv∈EGSu+SvSu×Sv2=824+54×5+2b−425+55×5+425+77×7+4b−425+85×8+4a−426+76×7+4a−426+86×8+2a27+97×9+2a+2b−428+88×8+4a+4b−828+98×9+15ab−18a−14b+1629+99×9NHGT3a,b=37200ab+15200a+911000b+2492500.


## 3. Conclusion

This paper is devoted to definingNGH, HG, and NHG indices, and the chemical applicability is studied for some physical and chemical properties of octane isomers using regression models including the recently introduced GH index. The GH index has a high negative correlation with acentric factor having r=0.987 with a residual standard error of 0.0059. The HG index has a high positive correlation with DHVAP having r=0.939 with a residual standard error of 0.136. The NGH index has a high negative correlation with acentric factor having r=0.877 with a residual standard error of 0.0176. The NHG index has a high positive correlation with acentric factor having r=0.877 with a residual standard error of 0.018. The applications of carbon nanotubes have considerably increased because of their excellent mechanical, thermal, and electrical properties. The novel indices introduced in this paper would be of great help to understand the physicochemical and biological properties of various compounds in addition to the existing degree-based indices.



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*Source: 3734185-2021-09-08.xml*