Chemical Applicability of Newly Introduced Topological Invariants and Their Relation with Polycyclic Compounds

In quantitative structure-property and structure-activity relationships studies, several graph invariants, namely, topological indices have been dened and studied due to their numerous applications in computer networks, biotechnology, and nanochemistry. Topological indices are numeric parameters that describe the biological, physical, and chemical properties depending on the structure and topology of dierent chemical compounds. In this article, we inaugurated some degree-based novel indices, namely, geometric-harmonic GHI, harmonic-geometric (HGI), neighborhood harmonic-geometric (NHGI), and neighborhood geometric-harmonic (NGHI) indices and veried their chemical applicability. Furthermore, an attempt is made to calculate analytical closed formulas for dierent variants of silicon carbides and analyze the obtained results graphically.


Introduction and Preliminaries
To reduce the consumption of time and cost during examining the biological, physical, and chemical properties of millions of newly invented nanomaterials, crystalline materials, and drugs, chemists study the quantitative structureproperty relationship (QSPR) and the quantitative structureactivity relationship (QSAR) [1][2][3][4]. Topological indices (TIs), numeric invariants describing properties of particular molecular structures, are used as a fundamental tool in the QSPR and QSAR. ese indices help in production of di erent chemicals with desired characteristics and estimation of physiochemical properties of existing compounds [5][6][7][8][9]. Another dominant bene t of the TIs is their e ectiveness during investigation of di erent aspects of chemical compounds and new drugs, which is the fundamental requirement of the medical sciences and industry. Consequently, studying and computing the behavior of the TIs of the molecular structures is a signi cant source to provide qualitative and quantitative information and therefore is one of the trends in modern research. Along with drug design, isomer discrimination, chemical documentation, and biological characterization, di erent applications of TIs in mathematical chemistry are described in [10][11][12][13][14][15].
Silicon carbides, which occur as incredibly abundant minerals in nature containing covalent bonds between carbon and silicon atoms, are biatomic compounds with tetrahedrally oriented layers of carbon and silicon atoms. Because of these strongly packed layers and short-length covalent bonds, SiC as consequences of nonoxidizing behavior, high melting points, high erosion resistance, thermal, and chemical stability, silicon carbides have vital industrial applications [16][17][18].
ese electrical properties and lowcost production methodologies give superiority to silicon carbides among other metals and semiconductors. Being one of the most extensively used wide bandgap materials, SiC performs a vital role in power industries by setting new principles in power savings as recti ers or switches in the system for data centers, wind turbines, solar cells, and electric vehicles along with high radiation and temperature-tolerant electronic appliances. Several papers have been devoted to the study of silicon and carbon-based structures, for details, see [19][20][21][22][23]. Some silicon carbides variants such as [a, b] and SiC 3 − I [a, b] are studied in this article. e main objective of our work is to inaugurate some degree and neighborhood degree-based novel indices, discuss the physical and chemical applicability of octane isomers using regression models, and compute defined indices for different variants of silicon carbides mentioned earlier.
is process provides an accurate estimation of physiochemical properties, i.e., boiling point, melting point, bond energy, and intermolecular forces. Particularly, these indices are helpful in predicting the properties of chemical compounds having polymerization such as silicon carbides, benzenoid hydrocarbons, and carbon nanotubes widely used in nanotech equipment. Now, we provide some preliminary concepts, while standard notations [24]  A number of topological indices have been introduced in the last decade such as the vertex connectivity index, edge connectivity index, geometric index, wiener index, harmonic index, Randic's molecular connectivity index, and face index [25][26][27][28][29][30]. e concept of neighborhood degrees was introduced by Chellali [31], and then, numerous articles have been published elaborating on these concepts [32][33][34]. Lately, Usha et al. [35] introduced the geometric-harmonic index combining geometric and harmonic indices, inspired by Furtula and Vukicevic [36] in designing the GHI. . (1) Shanmukha [37] et al. introduced three degree and neighborhood degree-based novel indices, namely, harmonicgeometric (HGI), neighborhood harmonic-geometric (NHGI), and neighborhood geometric-harmonic (NGHI) indices motivated by the above work. ey are defined as follows: To compute our results, we utilize the edge parcel technique, vertex segment strategy, graph hypothetical devices, degree checking strategy, expository strategies, the whole of degrees of the neighbor technique, and combinatorial techniques. In addition, MATLAB is utilized for mathematical calculations and verifications, whereas Maple is used for graphical analysis and plotting these scientific results.

Regression Models and Chemical Applicability
In this section, linear regression models of newly introduced topological indices and four physical properties of octane isomers as shown in Figure 1 are presented. ese models describe that physical properties, i.e., an acentric factor (AF), entropy (S), enthalpy of vaporization (HVAP), and standard enthalpy of vaporization (DHVAP) have excellent correlation with the GHI, NGHI, HGI, and NHGI. e linear regression models for S, AF, HVAP, and DHVAP are generated utilizing the method of least squares and given data in Table 1. e regression models for the GHI with mentioned physical properties are given as follows: In the above equations, the errors of regression coefficients are written within brackets. e residual standard (RS) error and the correlation coefficient for the regression models of GHI, HGI, NGHI,and NHGI with four physical properties are presented in Figures 2-5 and Tables 2-5 which clarify a significant association between these parameters.

Results for Si 2 C 3 − II[a, b]
Utilizing edge partition strategies, degree, and ev-degree based frequencies of different edges, two-dimensional molecular structure of Si 2 C 3 − II[a, b] is analyzed in Tables 6 and 7 whereas Figure 6 describes the unit molecular cell and Si 2 C 3 − II[4, 1] having 1 row and 4 molecules in each row.

Journal of Mathematics
By the definition of the degree-based geometric-harmonic index, harmonic-geometric index, and edge frequency of Si 2 C 3 − II[a, b] given in Table 6, we compute following results: By the definition of the ev-degree-based geometric harmonic index, harmonic geometric index, and edge frequency of Si 2 C 3 − II[a, b] given in Table 7, we compute following results: Journal of Mathematics

Results for SiC 3 − III[a, b]
Utilizing edge partition strategies, degree, and ev-degreebased frequencies of different edges, two-dimensional molecular structure of SiC 3 − III[a, b] is analyzed in Tables 8  and 9 whereas Figure 7 describes the unit molecular cell and SiC 3 − III [5,2] having 2 rows and 5 unit molecules in each row. By the definition of the degree-based geometric-harmonic index, harmonic-geometric index, and edge frequency of SiC 3 − III[a, b] given in Table 8, we compute following results: By the definition of the ev-degree-based geometric harmonic index, harmonic geometric index, and edge frequency of SiC 3 − III[a, b] given in Table 9, we compute following results: 12ab − 12a − 8b + 8

Results for Si 2 C 3 − III[a, b]
Utilizing edge partition strategies, degree, and ev-degree based frequencies of different edges, two-dimensional molecular structure of Si 2 C 3 − III [a, b] is analyzed in Tables 10  and 11, whereas Figure 8 describes the unit molecular cell and Si 2 C 3 − III [5,4] having 4 rows and 5 unit molecules in each row.By the definition of the degree-based geometricharmonic index, harmonic-geometric index, and edge frequency of Si 2 C 3 − III [a, b] given in Table 10, we compute following results: By the definition of the ev-degree-based geometricharmonic index, harmonic-geometric index, and edge frequency of Si 2 C 3 − III [a, b] given in Table 11, we compute following results: Journal of Mathematics

Conclusions
In this article, we discussed the regression models of newly introduced topological descriptors of the geometric-harmonic index, harmonic-geometric index, neighborhood geometric-harmonic index, and neighborhood harmonicgeometric index. We analyzed extraordinary correlation coefficients for these indices with different physical properties of octane isomers which represent the strong prediction abilities of the GHI, NGHI, HGI, and NHGI. Analytical values of these molecular variants are calculated to estimate physiochemical properties of silicon carbides. Our work allows researchers to calculate TIs for different polymeric compounds to elaborate on their chemical topologies and characteristics in the future. Also, we analyze the computed information and result graphically to understand their behavior with increasing molecular units in chemical structure of silicon carbides as shown in Figures 10  and 11.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Journal of Mathematics 15