Computation of Several Banhatti and Reven Invariants of Silicon Carbides

C and Si atoms, which were commonly found in diatomic layers. These layers combine to produce tetrahedral orientated C and Si atom molecules with a short bound length and high binding strength. Silicon carbide is the most extensively applicable material in Abstract Expressions for the molecular topological features of silicon carbide compounds are essential for quantitative structure-property and structure-activity interactions. Chemical Graph Theory is a subfield of computational chemistry that investigates topological indices of molecular networks that correlate well with the chemical characteristics of chemical compounds. In the modern age, topological indices are extremely important in the study of graph theory. Topological indices are critical tools for understanding the core topology of chemical structures while examining chemical substances. In this article, compute the first and second k-Banhatti index, modified first and second k-Banhatti index, first and second k-hyper Banhatti index, first and second hyper Revan indices, first Revan vertex index, and third Revan index for Silicon Carbide SiC4-II [p, q] for all values of p and q.


Introduction
Mathematical chemistry is a field of theoretical chemistry that uses mathematical approaches to discuss molecule structure without necessarily using quantum mechanics. Graph theory can be used to represent a chemical structure, with vertices representing atoms and edges representing chemical bonds. Chemical graph theory is a field of mathematical chemistry that bridges the gap between mathematics, chemistry, and graph theory to solve chemical issues mathematically. If there is a connection between any two vertices in a network, it is said to be connected 1 .
A molecular descriptor, also known as a topological graph index, is a mathematical formula that may be applied to any graph that describes a molecule structure. The topological index is used extensively in this field of research to investigate the topological features of various chemical structures.
The topological index is a numerical parameter associated with a chemical compound's molecular network. Topological indices are numerical values that describe the entire structure of a graph 2 . The topological indices are effective in predicting the physicochemical characteristics and bioactivity of the chemical compound. In mathematical chemistry, molecular descriptors serve an important role, particularly in the quantitative structure-property relationship (QSPR) and quantitative structureactivity relationship (QSAR) studies 3-6 .
Silicon Carbide (SiC) was the first substance to exhibit covalent bonds between C and Si atoms, which were commonly found in diatomic layers. These layers combine to produce tetrahedral orientated C and Si atom molecules with a short bound length and high binding strength. Silicon carbide is the most extensively applicable material in structural ceramics. Numerous applications have been facilitated by properties such as resistance to abrasion, Low density, low thermal expansion, high elastic modulus, high thermal conductivity, hardness, and corrosion, and most significantly, the maintenance of elastic resistance at temperatures up to 1650° C 7 . SiC finds numerous applications in a variety of industries due to its special properties, including abrasive and cutting tools, structural materials, automobile parts, foundry crucibles, electric systems, electronic circuit elements, power electronic devices, LEDs, astronomy, heating elements, nuclear fuel particles, jewelry, steel production, and quantum physics 8 . This is the backbone of the superior mechanical and chemical stability of SiC. Moissanite is an extremely rare mineral that contains SiC. In this paper, the particular isomer of the silicon carbides SiC4-II[p, q] is examined by using some of the vertex and edges degree-related topological indices.
The first T-index was the Wiener index, given by a famous chemist, H. Wiener 9 . It was a distancerelated index used to calculate the boiling point of the paraffin and named "path number" by Wiener. All the indices used in this article are vertex degreebased. The Zagreb indices are the oldest degreerelated indices introduced for the analysis of the pielectron energy of chemical compounds. Kulli proposed the first and second k-Banhatti indices by drawing inspiration from the work of the Zagreb indices. The modified form of B-indices is just the inverse of the classical B-indices. The hyper-Bindices are squares of B-indices, just like the hyper-Zagreb indices.

Fundamental Definitions
Suppose G is the graph that stands for all the chemical structures of the isomers of silicon carbide. All the chemical graphs used in this manuscript are simple, 2 dimensional, and planar. The nodes of the chemical networks are represented by "s" and "t" connected by the edges "e" to make the complex graphs. The distance between any two vertices "s" and "t" is the shortest path between them and is represented by ( , ). The number of edges attached to a vertex "s" is formed by its degree which is denoted by Φ( ). A new concept of the edge degree is proposed recently as; Φ(e) = Φ( ) + Φ( ) − 2. The maximum and minimum degree in a graph is represented by ∆(G) and δ(G).

 Kulli introduced the K-banhatti indices by
inspiring the work of Zagreb indices 10 . He analyzed various graphs such as cycle graphs, complete graphs, complete bipartite graphs, and regular graphs by using B-indices. The first k-Banhatti index 1 ( ) and second k-Banhatti index 1 ( ) are computed as: The B-induces is applicable to describe the properties of the path graphs and is used to compare the different graph operations 11 . A specific structure of the SiC namely SiC3-I [s, t] is analyzed by using the B-indices and Revan indices 12 . Nanomaterials are complex substances or materials that are synthesized and used on a tiny scale. The B-indices are used to explore the structure of different nanomaterials and famous Jahangir graphs 13,14 .
 Kulli suggested the first and second forms of modified B-indices. He applied these indices to study the behavior of connected graphs such as path, cycle, complete and bipartite graphs 15 . The modified first k-Banhatti index m 2(G) and second k-Banhatti index m 2(G) are determined as: Many topological indices are used to explore the different isomers of the silicon carbides, due to the requirement of silicon in the modern world. The modified indices are used to discuss the structure of Si2C3-III [s, t] and several classes of the Jahangir graph 16, 17 .
 Kulli suggested the hyper B-indices for the simple and connected graphs such as cycle, complete, and bipartite graph 18 . The formulas of the first k-hyper Banhatti index 1 ( ) and second k-hyper Banhatti index 2 ( ) are given below: All the forms of the B-indices are introduced by inspiring the different forms of Zagreb indices.
 The mathematical form of the first hyper Revan indices 1 ( ) and second hyper Revan indices 2 ( ) are written in the following way: and means that the vertex and vertex are adjacent in G. Revan indices have good correlation with not only the general mathematical graphs but also with the chemical graphs 23-26 . For more information related to graph theory see 27,28 .

Method and Strategies
Numerous methods, including vertex and edge partitioning, graph analytic tools, and combinatorial algorithms, are utilized to calculate the results. All of the degree-related indices are calculated by hand using a basic calculator, and the calculations are rechecked using MATLAB. Because two major variables are employed at the same time, their graphs are three-dimensional. Mathematica is used to create the 3D graphs, while Chem-Draw is used to construct the chemical structures of SiC4-II[p, q].

Structural Representation of SiC4-II[p, q]
The two-dimensional molecular structures of SiC4-II[p, q] are demonstrated in Fig.1

Vertex Partition
The whole vertex set of the graph SiC4 − II[p, q] is partitioned into three classes according to the degree.
The vertex sets with one, two, and three degrees are represented by V 1 , V 2 , and V 3 , respectively. Table. 1, shows the general form of all the vertex degrees with their frequencies and the general form of total vertices and edges is shown in Table. 2. These generalizations of the three parcels of vertices are done with the help of Matlab.  Edge Partition: The edge set of SiC4-II[p, q] is partitioned by using the methods described above. There are four different edge divisions of SiC4-II[p, q] that are described in Table 3

Numerical Analysis
This section discusses the numerical results of silicon carbide, SiC4-[p, q]. Numerical data are quantities that may be measured and arranged properly. Different values of p and q are used to examine the variance at various points. When the values of the input parameters (p, q) change, the values of the output (topological indices) vary as well. Table 4 displays the numerical values of all topological indices. Table 4

Graphical Expression of SiC4-II[p, q]
Fig. 3 demonstrates the variation in the outcomes of all topological indices. The variations of the parameters p and q are presented on the x-and yaxes, respectively, while the results of topological indices are given in three-dimensional space. All of the graphs in Fig. 3(a-e) were constructed using data from Table 4. Numerical analysis is an effective method for comparing data, and their graphs make the comparison extremely efficient. Graphs and charts reduce massive quantities of information into simple formats that express key ideas simply and efficiently. Graphs make information more accessible. This is particularly true when two or more groups of numbers are somehow connected. Graphs come in a variety of shapes and sizes, including bar charts, line graphs, area graphs, scatter plots, pie charts, pictographs, column charts, and bubble charts. For data analysis in this area, 3D space graphs are applied. The extreme values of the topological index are also depicted in these figures. These figures also show how the values of topological indices can change when p and q are changed. The relationship between topological indices and experimental measurements makes it simple to identify the characteristics of silicon structures SiC4-II[p, q].

Data Availability
In this article, no data were utilized.