Novel Degree-Based Topological Descriptors of Carbon Nanotubes

Department of Mathematics, Jain Institute of Technology, Davanagere-577003, Karnataka, India Department of Mathematics, Alliance School of Applied Mathematics, Alliance University, Bangalore-562106, Karnataka, India Department of Mathematics, Comsats University Islamabad, Lahore Campus, Lahore, Pakistan Department of Computer Science and Engineering, Bapuji Institute of Engineering and Technology, Davanagere-577004, Karnataka, India Department of Mathematics, University of Ghana, Legon, Ghana


Introduction
e 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. e topological index is a numerical descriptor, defines the graph topology of the molecule, and predicts an extensive range of molecular properties 5 [1][2][3][4][5][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][8][9][10][11].
Topological descriptors are frequently used in the discovery of drugs as they have rich datasets that give high predictive values. ese descriptors give the information depending on the arrangement of atoms and their bonds of a chemical compound. ey are studied for chemical compounds where, generally, the hydrogen atoms are suppressed. e originality of QSAR/QSPR models depends on physicochemical properties for chemical compounds with high degree of precision. ese models depend on various factors such as selecting the suitable compounds, suitable descriptors, and suitable algorithms or tools used in model development [12]. e QSAR/QSPR analysis is based on the data obtained by the numerical descriptors. ese 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.
e 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][14][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. e main use of alkanes is found in crude oil such as petroleum, cooking gas, pesticides, and drug synthesis. e 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).
A structure whose size is between the microscopic and molecular structure is referred to as a nanostructure. ere 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][18][19][20][21][22][23][24]. e nanostructures made of carbons with cylindrical shape are carbon nanotubes (CNTs).
ey have a similar structure to that of a fullerene and graphene except their cylindrical shape. e 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. ey are used as a grouting agent placed between the prosthesis and the bone as a part of their therapeutic use. e 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. ey 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. e 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.
ere 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. is 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. en, the atoms and bonds coincide with an imaginary graphene sheet. e length of the two atoms on opposite edges of the strip corresponds to circumference of the cylindrical graphene sheet [27][28][29].
e 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 C 4 C 8 (S), C 4 C 8 (R), and H-naphthalenic nanosheets Let G � (V, E) be a graph with a vertex set V(G) and an edge set E(G) such that |V(G)| � n and |E(G)| � m. For standard graph terminologies and notations, refer to [30,31].where (u, v) is an element of E(G), d u represents the degree of the vertex u, and S u 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: 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. ey are defined as follows:
From Table 3 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 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 correlation coefficient |r| � 0.873 with entropy, |r| � 0.695 with HVAP, and |r| � 0.778 with DHVAP.
From Table 5 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. In this section, GH, HG, NGH, and NHG indices of the C 4 C 8 (S) nanosheet are computed. e pattern of carbon atoms gives rise to two types of nanosheets, namely,

GH, NGH, HG, and NHG
e 2-dimensional nanosheet is represented by T 1 [a, b], where a and b are parameters ( Figure 6). In T 1 [a, b], C 4 acts as a square, while C 8 is an octagon in which a and b represent the column and row, respectively. Figure 7 depicts the type 1 − C 4 C 8 (S) nanosheet.
e number of vertices of the C 4 C 8 (S) nanosheet is 8ab, and the number of edges is 12ab − 2a − 2b.
e edge partition of the T 1 [a, b] nanosheet based on the degree of vertices is detailed in Table 6. an (a, b) Proof. Using Table 6, the definitions of GH and HG indices are as follows:           Journal of Chemistry 7 □ e edge partition of the T 1 [a, b] nanosheet based on the neighborhood degree of vertices is detailed in Table 7.

Conclusion
is paper is devoted to defining NGH, 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. e GH index has a high negative correlation with acentric factor having r � 0.987 with a residual standard error of 0.0059. e HG index has a high positive correlation with DHVAP having r � 0.939 with a residual standard error of 0.136. e NGH index has a high negative correlation with acentric factor having r � 0.877 with a residual standard error of 0.0176. e NHG index has a high positive correlation with acentric factor having r � 0.877 with a residual standard error of 0.018. e applications of carbon nanotubes have considerably increased because of their excellent mechanical, thermal, and electrical properties. e 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
e data used to support the findings of this study are cited at relevant places within the text as references.

Conflicts of Interest
e authors declare that they have no conflicts of interest.