Research Article  Open Access
M. C. Shanmukha, A. Usha, M. K. Siddiqui, K. C. Shilpa, A. AsareTuah, "Novel DegreeBased Topological Descriptors of Carbon Nanotubes", Journal of Chemistry, vol. 2021, Article ID 3734185, 15 pages, 2021. https://doi.org/10.1155/2021/3734185
Novel DegreeBased Topological Descriptors of Carbon Nanotubes
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 geometricharmonic, harmonicgeometric, and neighborhood harmonicgeometric indices. Also, the aforementioned three indices along with the geometricharmonic index are tested for physicochemical properties of octane isomers using linear regression models and computed for some carbon nanotubes.
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 properties 5[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]. Distancebased 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 treelike 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 Figure 1).
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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 bonerelated 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 singlewall 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 largescale 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 indices To discuss the physical and chemical applicability of octane isomers using regression models To compute defined indices for carbon nanotubes such as , , and Hnaphthalenic nanosheets
Let be a graph with a vertex set and an edge set such that and . For standard graph terminologies and notations, refer to [30, 31].where is an element of , represents the degree of the vertex , and represents the neighborhood degree of the vertex .
Definition 1. Recently, Usha et al. [32] defined the geometricharmonic (GH) index, inspired by Vukicevic and Furtula [33] in designing the GA index:Motivated by the above work, in this paper, an attempt is made to define three novel indices based on degree and neighborhood degree, namely, harmonicgeometric (HG), neighborhood geometricharmonic (NGH), and neighborhood harmonicgeometric (NHG) indices. They are defined as follows:
1.1. Chemical Applicability of , , , and Indices
In this section, a linear regression model of four physical properties is presented for , , , and indices. The physical properties such as entropy, acentric factor , enthalpy of vaporization , and standard enthalpy of vaporization of octane isomers have shown good correlation with the indices considered in the study. The , , , and indices are tested for the octane isomers’ database available at https://www.moleculardescriptors.eu/dataset.htm. The , , , and indices are computed and tabulated in columns 6, 7, 8, and 9 of Table 1.

Using the method of least squares, the linear regression models for S, AF, HVAP, and DHVAP are fitted using the data of Table 1.
The fitted models for the index are
The fitted models for the index are
The fitted models for the index are
The fitted models for the index are
Note: in equations (3)–(18), the errors of the regression coefficients are represented within brackets.
Tables 2–5 and Figures 2–5 show the correlation coefficient and residual standard error for the regression models of four physical properties with , , , and indices.




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From Table 2 and Figure 2, it is obvious that the index highly correlates with the acentric factor and the correlation coefficient . Also, the index has good correlation coefficient with entropy, with , and with .
From Table 3 and Figure 3, it is noticed that the index highly correlates with and the correlation coefficient . Also, the index has good correlation coefficient with entropy, with the acentric factor, and with .
From Table 4 and Figure 4, it is clear that the index highly correlates with the acentric factor and the correlation coefficient . Also, the index has good correlation coefficient with entropy, with , and with .
From Table 5 and Figure 5, it is clear that the index highly correlates with the acentric factor and the correlation coefficient . Also, the index has good correlation coefficient with entropy, with , and with .
2. GH, NGH, HG, and NHG Indices of , , and HNaphthalenic Nanosheets
2.1. Results for the Nanosheet
The alternating pattern of 4 carbon atoms forming squares and 8 carbon atoms forming octagons constitutes the nanosheet.
In this section, , , , and indices of the nanosheet are computed. The pattern of carbon atoms gives rise to two types of nanosheets, namely, and . The 2dimensional nanosheet is represented by , where and are parameters (Figure 6). In , acts as a square, while is an octagon in which and represent the column and row, respectively. Figure 7 depicts the type nanosheet. The number of vertices of the nanosheet is , and the number of edges is .
The edge partition of the nanosheet based on the degree of vertices is detailed in Table 6.

Theorem 1. Let be an dimensional nanosheet; then, GH and HG indices are equal to
Proof. Using Table 6, the definitions of GH and HG indices are as follows:
The edge partition of the nanosheet based on the neighborhood degree of vertices is detailed in Table 7.

Theorem 2. Let be an dimensional nanosheet; then, NGH and NHG indices are equal to
Proof. Using Table 7, the definitions of NGH and NHG indices are as follows:
2.2. Results for the 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 as . The 2dimensional lattice of the nanosheet, where and are parameters, is shown in Figure 8. Figure 9 shows the type nanosheet. In the following theorem, , , , and indices of this nanosheet are computed. The number of vertices of the type2 structure is , and the number of edges is .
The edge partition of the nanosheet for degreebased vertices is detailed in Table 8.

Theorem 3. Let be an dimensional nanosheet; then, GH and HG indices are equal to
Proof. Using Table 8, the definitions of and indices are as follows:
The edge partition of the nanosheet based on the neighborhood degree of vertices is detailed in Table 9.

Theorem 4. Let be an dimensional nanosheet; then, and indices are equal to
Proof. Using Table 9, the definitions of and indices are as follows:
2.3. Results for the HNaphthalenic Nanosheet
Carbon atoms bonded in the form of a hexagonal structure constitute carbon nanotubes. They are pericondensed benzenoids which mean three or more rings share the same atoms. HNaphthalenic nanosheet is constituted by the alternating sequence of squares , hexagons , and octagons and is represented as , where and are the parameters. The number of vertices of the Hnaphthalenic nanosheet is , and the edges are . The , , , and indices of this nanosheet are computed; see Figure 10.
The edge partition of the nanosheet based on the degree of vertices is detailed in Table 10.

Theorem 5. Let be an dimensional nanosheet; then, and indices are equal to
Proof. Using Table 10, the definitions of and indices are as follows:
The edge partition of the nanosheet based on the neighborhood degree of vertices is detailed in Table 11.

Theorem 6. Let be an dimensional nanosheet; then, and indices are equal to
Proof. Using Table 11, the definitions of and indices are as follows:
3. Conclusion
This paper is devoted to defining , , and indices, and the chemical applicability is studied for some physical and chemical properties of octane isomers using regression models including the recently introduced index. The index has a high negative correlation with acentric factor having with a residual standard error of 0.0059. The index has a high positive correlation with having with a residual standard error of 0.136. The index has a high negative correlation with acentric factor having with a residual standard error of 0.0176. The index has a high positive correlation with acentric factor having 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 degreebased indices.
Data Availability
The data used to support the findings of this study are cited at relevant places within the text as references.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this work.
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Copyright © 2021 M. C. Shanmukha et al. This 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.