Chemical applicability and computation of K-Banhatti indices for benzenoid hydrocarbons and triazine-based covalent organic frameworks

The novel applications in chemistry include the mathematical models of molecular structure of the compounds which has numerous findings in this area that refers to mathematical chemistry. Topological descriptors play a major role in QSAR/QSPR studies that analyses the biological and physicochemical properties of the compounds. In the recent times, a new type of topological descriptors are proposed, called K-Banhatti indices. In this study the chemical applicability of K-Banhatti indices are examined for benzenoid hydrocarbons (derivatives of benzene). These indices have shown remarkable results through the study of statistical analysis. Subsequently, triazine-based covalent organic frameworks (CoF’s) are studied for which \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_1(G)$$\end{document}B1(G), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$B_2(G)$$\end{document}B2(G), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$HB_1(G)$$\end{document}HB1(G), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$HB_2(G)$$\end{document}HB2(G), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^mB_1(G)$$\end{document}mB1(G), \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${}^mB_2(G)$$\end{document}mB2(G), and HB(G) of a graph G are computed.


Chemical applicability of K-Banhatti indices
This section concentrates on framing the linear regression model for the properties such as boiling point (BP), enthalpy (E), π -electron energy ( π-ele), and molecular weight (MW) of benzene derivatives 27,28 for the consid- ered indices using Tables 1 and 2. It is noticed from Tables 3, 4, 5, 6, 7, 8 and 9 that the regression model of statistical parameters show significant values and the coefficient of correlation R with the above four properties show high positive correlation (also see Fig. 2).It is evident from Table 10 that, B 1 (G) , B 2 (G) , HB 1 (G) , HB 2 (G) , m B 1 (G) , HB(G) are highly correlated with the π -electron energy, while m B 2 (G) highly correlated with molecular weight.4. The linear regression models for the second K-hyper Banhatti index(HB 2 ) 5. The linear regression models for the modified first K-Banhatti index( m B 1 ) 6.The linear regression models for the modified second K-Banhatti index( m B 2 ) 7. The linear regression models for the harmonic K-Banhatti index(HB)

Triazine based covalent organic frameworks (CoF's)
The chemical systems that possess discrete number of molecules refers to supra-molecular chemistry.The spatial arrangement of the molecules is responsible for the strength of the forces between them may be weak or strong.These forces may be due to intermolecular, hydrogen bonding, electrostatic charge, and covalent bonding.The  This work pinpoints on the supramolecular structure called triazine-based covalent organic frameworks (Fig. 3).To understand better about the chemical and biological properties of a chemical compound, graph theory uses a very useful tool called topological index.These indices help the chemists to derive information about the compound that may be in turn useful in drug design or drug delivery.Chemical graph theory is a combination of chemistry and mathematics in which the compound under the study will be modelled as a graph and the information about its atoms and their bonds are better understood.Chemical graph theory is the result of the strong linkages between both the subjects which have the outcomes as various significant investigations [29][30][31][32][33] .
Biologically significant organic molecules have a new dimension as triazines act as the building blocks used in its design.Triazines and its derivatives have varied applications in antifungal, anticancer, antiviral, cardiotonic, anti-HIV, analgesic, etc., with fine tuned electronic properties.The goal of scientific researchers is to apply their theoretical research in industrial applications so that it is useful for humankind.The objective is to make the products scalable and satisfy excellent properties obtained from the experiments at a reasonable cost and long-term stability.
Covalent organic frameworks (CoF's) have attracted various researchers across the globe because of its excellent properties such as adsorption, chemo-sensing, energy storage and production.As the CoFs and their applications are found in industries, many research achievements have come to light recently [34][35][36][37] .CoF's may be classified into boron-containing, triazine-based, imide-linked and imine-based due to swift increasing requirements in various fields.We focus on the second category of CoF's, i.e., triazine-based in this study.In 2008, Thomas et al. 38 , prepared triazine based CoF, by cyclotrimerization of nitrile building units at 400 • C in the presence of ZnCl 2 .There was destruction of ordered structure, despite the harsh conditions during the preparation which included high reaction temperature and purification in acid solution.However, few triazine-based CoF's show crystallinity, and these building blocks were unable to adapt to harsh temperatures.Later, triazine based CoF's (CTFs) were synthesized by the condensation reaction of aldehydes and amidines 39 .
The distinguishing physical and chemical features of CoF's have led to the plethora of applications in the industries.In 2011, Ding et al. reported first set of CoF's that are useful in the field of catalysis 40 .It was noted that 2-dimensional CoF's acts as catalyst in different reactions that include nitrophenol reduction, water oxidation, in reducing CO 2 to CO etc. CoF's are used tackle the problem of excessive CO 2 emissions as they are the principal reason for greenhouse effect.It is mainly due to the expansion of population and the development of industries.Aqueous alkanolamine is proposed to implement the CO 2 emissions.To control the CO 2 emissions, new materials are to be developed with high performance in which CoF's play a significant role.Also, CoF plays an important role in energy storage 41,42 .
Augustine et al. 43,44 theoretically examined triazine-based covalent-organic frameworks (CoF's) using vertex and edge partition for degree-based and neighborhood degree-based topological indices.Additionally, the degree-based and neighborhood degree-based entropy measures for the results are given.The graph theoretical approach is used to compare the outcomes with obtained results.In this section, based on the previous work of Tony Augustine where, where,  .Also, the edge set of hexagonal TriCF structure is classified into two edge partitions depending on the vertex degrees are given by (see 44 ) such that We have by the definition of first K-Banhatti index B 1 (G) is given by similarly, and HB(G) = 198n 2 + 10n 5 .
where,   Table 11 shows the numerical comparison of K-Banhatti indices for linear chain TriCF structure which linearly increases as n increase.Table 12 shows the variation of the indices under the study for parallelogram TriCF which increases as n, m increase.Finally, Table 13 shows the increase in the indices as n increase.

Conclusion
The molecules are modelled, and their physicochemical and biological properties are predicted using the Quantitative structure-property relationships (QSPR) studies.Topological index is a significant tool used by QSPR studies in encoding the information of a molecule.There are a bunch of topological indices which are of significant importance in the properties of the compounds based on its algorithm defined.
, and HB(G) for benzenoid hydrocarbons of a graph G are examined and it is observed that the considered indices (molecular descriptors) showed good predictive potential.Benzenoid hydrocarbons have numerous applications because of its unique physical and chemical properties.Some of them include paint thinners, laminates, cement, in medicine for curing bacterial infections, mosquito repellents, cosmetics, toothpaste, detergents, and a dyeing agent.
Also the above said indices are computed for triazine-based covalent organic frameworks (CoF's).Triazine has wide applications in industries, where one of the famous forms being melamine.It is used in kitchen appliances and carpentry.Another form of triazine is cyanuric chloride that are used in reactive dyes and herbicides.It has several applications in oil, petroleum and gas processing industries.They are used to remove harmful hydrogen sulphide gas and other species from fluid streams in infrastructure.As the chemical compound, triazine has many applications especially in industries, the work can be extended for other indices using graph operators and see the variation.Also, it has applications in medical field, attracting the pharmacists and chemists in the usage of drug design and delivery.

Figures 4 ,
Figures 4, 5 and 6 showcase the structures of linear chain, Parallelogram and Hexagonal triazine -based covalent oraganic frame works (TriCF) for which the edge and vertex partitions are determined and hence the various forms of K-Banhatti indices are computed.Figures 7, 8 and 9 represents the graphical comparison of K-Banhatti indices for linear chain TriCF ( n ∈ {1, 2, 3, . . ., 10} ), Parallelogram TriCF ( n ∈ {1, 2, 3, . . ., 10} ) and Hexagonal TriCF ( n ∈ {1, 2, 3, . . ., 10} ) respectively.The figures show that the first K-Banhatti index(B 1 ) has more value compared with other K-Banhatti indices while m B 1 and m B 2 showcase the least values and hence it is very close to the x-axis in all the graphs for all triazine-based covalent organic frame works (CoF's).Table11shows the numerical comparison of K-Banhatti indices for linear chain TriCF structure which linearly increases as n increase.Table12shows the variation of the indices under the study for parallelogram TriCF which increases as n, m increase.Finally, Table13shows the increase in the indices as n increase.

Table 2 .
The values of K-Banhatti indices for benzenoid hydrocarbons.

Table 3 .
The regression model of statistical parameters for B 1 .

Table 4 .
The regression model of statistical parameters for B 2 .

Table 5 .
The regression model of statistical parameters for HB 1 .

Table 6 .
The regression model of statistical parameters for HB 2 .

Table 7 .
The regression model of statistical parameters for m B 1 .

Table 9 .
The regression model of statistical parameters for HB.

Table 10 .
Correlation coefficients (R) between physicochemical properties of benzene derivatives, and K-Banhatti indices.

Table 11 .
The comparison of K-Banhatti indices for linear chain of TriCF structure.

Table 13 .
The comparison of K-Banhatti indices for hexagonal TriCF structure.