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Application of Molecular Topological Descriptors in Chemistry

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Volume 2021 |Article ID 3734185 | https://doi.org/10.1155/2021/3734185

M. C. Shanmukha, A. Usha, M. K. Siddiqui, K. C. Shilpa, A. Asare-Tuah, "Novel Degree-Based Topological Descriptors of Carbon Nanotubes", Journal of Chemistry, vol. 2021, Article ID 3734185, 15 pages, 2021. https://doi.org/10.1155/2021/3734185

Novel Degree-Based Topological Descriptors of Carbon Nanotubes

Academic Editor: Ajaya Kumar Singh
Received27 Jul 2021
Accepted27 Aug 2021
Published08 Sep 2021

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.

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[16].

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 [711].

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 [1315]. 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 Figure 1).

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 [1724]. 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 [2729].

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 as , , and H-naphthalenic 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 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. 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.


AlkaneSAFHVAPDHVAPGHNGHHGNHG

n-Octane111.7000.39873.1909.91524.42384.4962.1930.679
2-Methylheptane109.8000.37870.3009.48427.17398.7461.9620.573
3-Methylheptane111.3000.37171.3009.52127.954107.4752.0580.568
4-Methylheptane109.3000.37270.9109.48327.954113.3562.0580.600
3-Ethylhexane109.4000.36271.7009.47628.735131.9412.1540.476
2,2-Dimethylhexane103.4000.33967.7008.91533.607122.9701.6890.471
2,3-Dimethylhexane108.0000.34870.2009.27231.637110.6221.8620.498
2,4-Dimethylhexane107.0000.34468.5009.02930.885123.2361.8270.474
2,5-Dimethylhexane105.7000.35768.6009.05132.248133.2421.7310.469
3,3-Dimethylhexane104.7000.32368.5008.97335.213151.3981.8290.449
3,4-Dimethylhexane106.6000.34070.2009.31632.418117.7011.9580.578
2-Methyl-3-ethylpentane106.1000.33269.7009.20932.418176.1111.9580.342
3-Methyl-3-ethylpentane101.5000.30769.3009.08136.820149.2221.9680.377
2,2,3-Trimethylpentane101.3000.30167.3008.82638.834164.3661.6060.338
2,2,4-Trimethylpentane104.1000.30564.8708.40236.537155.8741.4590.358
2,3,3-Trimethylpentane102.1000.29368.1008.89739.659141.6571.6490.462
2,3,4-Trimethylpentane102.4000.31768.3709.01435.321168.7971.6660.390
2,2,3,3-Trimethylbutane93.0600.25566.2008.41046.000223.6201.2630.227

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 25 and Figures 25 show the correlation coefficient and residual standard error for the regression models of four physical properties with , , , and indices.


Physical propertiesValue of the correlation coefficientResidual standard error

Entropy−0.9681.17
Acentric factor−0.9870.0059
HVAP−0.8151.21
DHVAP−0.8850.184


Physical propertiesValue of the correlation coefficientResidual standard error

Entropy0.852.448
Acentric factor0.8330.020
HVAP0.9350.74
DHVAP0.9390.136


Physical propertiesValue of the correlation coefficientResidual standard error

Entropy−0.8732.274
Acentric factor−0.8770.0176
HVAP−0.6951.502
DHVAP−0.7780.248


Physical propertiesValue of the correlation coefficientResidual standard error

Entropy0.8522.436
Acentric factor0.8770.018
HVAP0.7771.315
DHVAP0.8430.213

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 H-Naphthalenic 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 2-dimensional 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.


with Number of edges


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.


with Number of edges

4
8

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 2-dimensional 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 type-2 structure is , and the number of edges is .

The edge partition of the nanosheet for degree-based vertices is detailed in Table 8.


with Number of edges

4

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.


with Number of edges

4
8

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 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 squares , hexagons , and octagons and is represented as , where and are the parameters. The number of vertices of the H-naphthalenic 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.


with Number of edges


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.


with Number of edges

8
4

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 degree-based 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.

References

  1. D. Bonchev, Information Theoretic Indices for Characterization of Chemical Structures, Research Studies Press, 1983.
  2. B. Furtula and I. Gutman, “A forgotten topological index,” Journal of Mathematical Chemistry, vol. 53, no. 4, pp. 1184–1190, 2015. View at: Publisher Site | Google Scholar
  3. I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, Germany, 2012.
  4. I. Gutman and N. Trinajstić, “Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons,” Chemical Physics Letters, vol. 17, no. 4, pp. 535–538, 1972. View at: Publisher Site | Google Scholar
  5. R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley VCH, Weinheim, Germany, 2000.
  6. N. Trinajstic, Chemical Graph Theory, CRC Press, Boca Raton, FL, USA, 1992.
  7. S. Hayat, M. Imran, and J. B. Liu, “Correlation between the Esrada index and -electron energies for benzenoid hydrocarbons with applications to boron nanotubes,” International Journal of Quantum Chemistry, vol. 119, pp. 1–13, 2019. View at: Publisher Site | Google Scholar
  8. M. Randic, “Quantitative structure-property relationship. Boiling points of planar benzenoids,” New Journal of Chemistry, vol. 20, pp. 1001–1009, 1996. View at: Google Scholar
  9. M. C. Shanmukha, A. Usha, N. S. Basavarajappa, and K. C. Shilpa, “Novel Neighbourhood redefined First and Second Zagreb indices on Carborundum Structures,” Journal of Applied Mathematics and Computing, vol. 66, pp. 1–14, 2020. View at: Google Scholar
  10. M. C. Shanmukha, N. S. Basavarajappa, K. C. Shilpa, and A. Usha, “Degree-based topological indices on anticancer drugs with QSPR analysis,” Heliyon, vol. 6, Article ID e04235, 2020. View at: Google Scholar
  11. S. A. K. Kirmani, P. Ali, F. Azam, and P. A. Alvi, “Topological indices and QSPR/QSAR analysis of some antiviral drugs being investigated for the treatment of COVID-19 patients,” International Journal of Quantum Chemistry, vol. 121, pp. 1–22, 2021. View at: Publisher Site | Google Scholar
  12. K. Roy, “Topological descriptors in drug design and modeling studies,” Molecular Diversity, vol. 8, no. 4, pp. 321–323, 2004. View at: Publisher Site | Google Scholar
  13. A. T. Balaban and J. Devillers, Topological Indices and Related Descriptors in QSAR and QSPR, CRC Press, London, UK, 2014.
  14. J. C. Dearden, Advances in QSAR Modeling, Springer, Cham, Switzerland, 2017. View at: Publisher Site
  15. E. Estrada, L. Torres, L. Roriguez, and I. Gutman, “An atom-bond connectivity index: modelling the enthalpy of formation of alkanes,” Indian Journal of Chemistry, vol. 37, pp. 849–855, 1998. View at: Google Scholar
  16. H. Wiener, “Structural determination of parafin boiling points,” Journal of the American Chemical Society, vol. 69, no. 1, pp. 17–20, 1947. View at: Publisher Site | Google Scholar
  17. B. Basavangouda and S. Policepatil, “Chemical applicability of Gourava and hyper-Gourava indices,” Nanosystems: Physics, Chemistry, Mathematics, vol. 12, pp. 142–150, 2021. View at: Google Scholar
  18. P. S. R. Harisha and V. Lokesha, “M-Polynomial of valency-based topological indices of operators on titania nanotubes networks,” Advances and Applications in Mathematical Sciences, vol. 20, pp. 1493–1508, 2021. View at: Google Scholar
  19. P. S. Hemavathi, V. Lokesha, M. Manjunath, P. S. KReddy, and R. Shruti, “Topological aspects boron triangular nanotube and boron- nanotube,” Vladikavkaz Mathematical Journal, vol. 22, pp. 66–77, 2020. View at: Google Scholar
  20. V. Lokesha, T. Deepika, P. S. Ranjini, and I. N. Cangul, “Operations of nanostructures via SDD, ABC 4 and GA 5 indices,” Applied Mathematics and Nonlinear Sciences, vol. 2, no. 1, pp. 173–180, 2017. View at: Publisher Site | Google Scholar
  21. V. Lokesha, R. Shruti, P. S. Ranjini, and S. Cevik, “On certain topological indices of nanostructures using and perators,” Communications, vol. 67, pp. 178–187, 2018. View at: Google Scholar
  22. S. Mondal, A. Bhosale, N. De, and A. Pal, “Topological properties of some nanostructures,” Nanosystems: Physics, Chemistry, Mathematics, vol. 11, no. 1, pp. 14–24, 2020. View at: Publisher Site | Google Scholar
  23. M. F. Nadeem, S. Zafar, and Z. Zahid, “On topological properties of the line graphs of subdivision graphs of certain nanostructures,” Applied Mathematics and Computation, vol. 273, pp. 125–130, 2016. View at: Publisher Site | Google Scholar
  24. Y. Shanthakumari, P. Siva Kota Reddy, V. Lokesha, and P. S. Hemavathi, “Topological aspects boron triangular nanotube and boron- nanotube-II, south east asian,” Journal of Mathematics and Mathematical Sciences (SEAJMMS), vol. 16, pp. 145–156, 2020. View at: Google Scholar
  25. S. Iijima, “Synthesis of Carbon Nanotubes,” Nature, vol. 354, pp. 56–58, 1991. View at: Publisher Site | Google Scholar
  26. S. Hayat, M. K. Shafiq, A. Khan et al., “On topological properties of 2-dimensional lattices of carbon nanotubes,” Journal of Computational and Theoretical Nanoscience, vol. 13, no. 10, pp. 6606–6615, 2016. View at: Publisher Site | Google Scholar
  27. A. Ali, W. Nazeer, M. Munir, and M. K. Shin, “M-polynomials and topological indices of zigzag and rhombic benzenoid systems,” Open Chemistry, vol. 16, pp. 73–78, 2017. View at: Google Scholar
  28. A. Q. Baig, M. Imran, and H. Ali, “Computing omega, sadhana and PI polynomials of benzenoid carbon nanotubes,” Optoelectronics and advanced materials-rapid communications, vol. 9, pp. 248–255, 2015. View at: Google Scholar
  29. M. Nadeem, A. Yousaf, and A. Razaq, “Certain polynomials and related topological indices for the series of benzenoid graphs,” Scientific Reports, vol. 9, pp. 1–6, 2019. View at: Publisher Site | Google Scholar
  30. F. Harary, Graph Theory, Addison-Wesely, Reading Mass, Boston, MA, USA, 1969.
  31. V. R. Kulli, College Graph Theory, Vishwa International Publications, Gulbarga, India, 2012.
  32. A. Usha, M. C. Shanmukha, K. N. Anil Kumar, and K. C. Shilpa, “Comparision of novel index with geometric-arithmetic and sum-connectivity indices,” Journal of Mathematical and Computational Science, vol. 11, pp. 5344–5360, 2021. View at: Google Scholar
  33. D. Vukicevic and B. Furtula, “Topological index based on the ratios of geometric and arithmetical means of end-vertex degrees of edges,” Journal of Mathematical Chemistry, vol. 46, pp. 1369–1376, 2009. View at: Google Scholar

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.


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