Bi-Distance Approach to Determine the Topological Invariants of Silicon Carbide

: The use of silicon carbide is increasing significantly in the fields of research and technology. Topological indices enable data gathering on algebraic graphs and provide a mathematical framework for analyzing the chemical structural characteristics. In this paper, well-known degree-based topological indices are used to analyze the chemical structures of silicon carbides. To evaluate the features of various chemical or non-chemical networks, a variety of topological indices are defined. In this paper, a new concept related to the degree of the graph called "bi-distance" is introduced, which is used to calculate all the additive as well as multiplicative degree-based indices for the isomer of silicon carbide, Si 2 C 3 -1[t, h]. The term "bi-distance" is derived from the concepts of degree and distance in such a way that second distance can be used to calculate degree-based topological indices.


Introduction:
Leonhard Euler (1702-1782) originated the term "graph" in graph theory in the eighteenth century. He was a mathematician from Switzerland. He used graph manipulation to solve Konigsberg Bridge problems 1 .
The subject of mathematics termed "graph theory" deals with structures of vertices represented by a series. Graph theory has evolved into an important field of mathematical research with relevance in chemistry, operations research, the social sciences, and computer science. All of the graphs in this paper are simple, connected, and planar. A graph is formed of vertices, nodes, or points that are connected by edges, arcs, or lines. "Graph theory" is the word used in mathematics to describe the analysis of graphs, which are mathematical patterns used to express pair-wise relationships among variables.
Graphs are one of the major aspects of discrete mathematics, and they have diverse applications in our daily life. The implementation of graph theory can be seen in nano-chemistry, computer networks, Google maps, and molecular graphs. Chemical graph theory is a sub-field of mathematical chemistry that uses graph theory to mathematically model chemical structures. It integrates chemistry and graph theory to investigate the physical and chemical properties of substances in more depth 2 .
According to the IUPAC terminology 3 , a "topological index" is a numerical number associated with a chemical composition that is used to correlate the chemical structure with numerous physical attributes, chemical reactivity, or bioactivities. In the recent two decades, it has become very widespread to investigate the physicochemical and structural features of molecular graphs, which are vital to chemical engineering and pharmaceutical research, using graph-theoretical methods. The T-indices are a method for calculating network properties. The usefulness of topological indices is determined by the correlation between experimental and estimated values. The distance-related indices in network theory and the degree-based indices in the chemical and pharmaceutical industries have both been shown to be extremely efficient. T-indices provide a simple and theoretical method to obtain in-depth knowledge about drugs by estimating the structural characteristics of a series of pharmaceuticals [4][5][6][7] .
Topological indices are quantitative measurements that do not depend on the geometry of the graph. Topological indices are used in the establishment of quantitative structure-activity relationships, which connect the bioactivity or other features of molecules using their chemical composition 8 .
The heat of formation, the heat of evaporation, density, and pressure are a few more factors that may be evaluated using these graph descriptors. For understanding chemical processes like evaporation, heating, and flashpoints, topological indices are significant. It is a numerical number that describes the structure of the chemical graph in other aspects. Many researchers are interested in this advanced method for estimating compound features without conducting any experiments. Due to their great significance, Tindices are classified into various categories, such as degree-based, eccentricity-based, distance-based, and ev-degree-based.
Sardara recently investigated the characteristics of a certain isomer of silicon carbide and presented double silicon graphs 9 . He used the additive and multiplicative topological versions of topology to learn more about SiC. Pan used the degree-related Banhatti and Revan indices 10 to investigate the two-dimensional structures of a particular class of silicon 2 3 − [ , ]. The silicon material has greatly inspired and motivated research interest due to its extraordinary mechanical, optical, and electrical capabilities. Xing-Long successfully used the entropy technique to analyze the shape and structure of silicon carbide 11 . Sadia Akhter 12 employed a novel topological approach to study the structure of two silicon isomers in 2019.

Preliminaries
A graph = ( , E) is composed of a collection of links and a collection of vertices V. The term ժ( , ) represents the distance between any two vertices in the simple and connected graph G. The degree of a vertex in G is represented by ժ( ), and the degree of a vertex in G by ժ( ). All the graphs used in this article are simple (without multiple edges and loops), connected, and planar (without edge crossings) 13 . The indices are degreebased, both additive and multiplicative indices.
 First and Second Zagreb Index: The degree-based Zagreb indices were proposed by Gutman and Trinajstic in 1972 14 . The first and second Zagreb indices, M1 and Z2, respectively, are equal to the sum of the squares of the degrees of the vertices and the products of the degrees of the pairs of adjacent vertices in the molecular graph. The formulae of these indices are: The multiplicative form of the Zagreb indices is used to investigate the other silicon carbide isomers 18, 19 . The properties of unicycle graphs and graphs with bridges are investigated using multiplicative M-indices 20 . The Isaac graph is a very important graph; hence, these polynomials are a useful tool for studying Isaac graphs in depth 22 .
The Hyper-Zagreb index is a modified Zagreb index that was proposed by Shirdel, Rezapour, and Sayadi 23 and publicized in 2013. They explain the newly proposed index for the cartesian product, composition, join, and disjunction of graphs.
Hyper-Zagreb index is denoted by HM(G) and computed as: The modified versions of the Zagreb indices are inspired by the usefulness of the classical Zagreb indices. The M2(G) is determined as:  Reduced Second Zagreb Index: Furtula, Gutman, and Ediz 24 investigated the difference between Zagreb indices and discovered that it is closely connected to the vertex-degree-based invariant known as the reduced second Zagreb index, written as: To study further information about all Zagreb indices see 25 .  Atom Bond Connectivity Index: In 1998, Ernesto Estrada and Fernando Torres proposed the ABC-index after being inspired by Milan Randic's work 26 . It is used to simulate the thermal properties of organic substances.
́ Index: Millan Randić suggested the first degree-based index in 1975 to understand the branching structure of carbon atoms in organic compounds 27 . The Randić The index is defined as: ́ Connectivity Index: In 1998, Bollob and Erdos generalized Milan's index by replacing 1\2 with any general number 28 .
 Reciprocal ́Index: Favaron, Maheó and Saclé were the first to introduce this form of R-index. The mathematical formula of RR-index is: ́ Index: It is the advanced form of the R-index and its mathematical definition is:  Geometric Arithmetic Index: Vukicevic and Furtula suggested the GAindex in 2009 29 which is described as:  Forgotten Index: Gutman and Furtula published this index in 2015 30 , and it is represented as F(G) which is described as:  General Sum Connectivity Index: Zhou and Trinajstić suggested the general form of the sum connectivity index. The ∝ (G)-index is mathematically written as:  Symmetric Division Index: The degree based symmetric division index was introduced by Vukicević and Furtula.
This index is very effective to predict the total surface area for poly-chloro-biphenyls.  Harmonic Index: In graph theory, Siemion Fajtlowicz created a computer program that generates conjectures automatically in 1990. He discovered a vertex degree-based quantity while working on this project. Zhang later retrieved that unknown quantity (in 2012) and termed it harmonic index 31 .
It's written like this: This article deals with the degree-based topological indices of silicon carbides. Silicon carbide is a highly unusual structure since it has various properties such as low density, strong strength, good high-temperature strength, low thermal expansion, and high thermal conductivity. indices of the different isomers of SiC 32-35 . Graph theory defines broad and advanced ideas to facilitate the understanding of many problems in different fields 36-39 .

Comparison of Single and Bi-Distance Edge Based Indices
All the classical degree-based TIs are singledistance. The word "single distance" does not appear in these indices. Due to the usefulness of TIs in real life, many approaches have been introduced, like distance-based, eccentricity-based, metricbased, additive type, multiplicative type, etc. The bi-distance strategy is a new method that is suggested in this article for determining these indices. A bi-distance edge is formed by combining two edges. The bi-distance concept is also used for finding the wiener index. The Wiener index just is concerned with all types of distances, but here the special term "two-distance" is proposed. In this section, the methyl-heptane single-distance edge and bi-distance edge partitions are examined.

 Single-distance Edge Partitions:
The methyl-heptane structure's edges are separated into several groups using the method for edge separation discussed above. Four distinct methylheptane edge bundles are presented in Table 1. The parcel 1 has 2 edges, where ժ = 1 and ժ = 3. The bundle 2 consist of only one edge, where ժ = 3 and ժ = 2. The pack 3 has 3 edges, where ժ = 2 and ժ = 2. The fourth edge bundle 4 is made up of 1 edge, where ժ = 2 and ժ = 1. All the calculations related to edge separation are given in Table 1.
The M1(G) -index for methyl heptane is determined by using the formula and the data given in Table 1.

 Bi-distance Edge Partitions:
The edge partition technique is applied to split the bi-distance edges of methyl-heptane into four packets given in Table 2. The parcel 1 composed of 3 edges, where ժ = 1 and ժ = 2. The bundle 2 is made by 1 edge, where ժ = 3 and ժ = 2. The third edge bundle has two edges, while the fourth edge parcel has just one edge. Table 2. Bi-distance edge partition of Methyl-heptane The M1(G)-index is determined as:

Methods
There are several methods and techniques for obtaining results, such as vertex degree, edge partitioning, graph analytical approaches, and numerical comparison of the results. Different software is used in this article. For computations and rechecking, MATLAB is really beneficial software. Software like Mathematica is used for 2D and 3D graphs that are used to represent the comparison of topological indices. ChemSketch was used for the structural graphs of Si2C3-I [t, h]. ChemDraw can also be used for drawing chemical structures in an easy way. compounds are explored due to the use and great importance of SiC in the modern world. Silicon has decreased the size of electronic devices and enhanced their quality. Silicon carbide is a crystalline combination of silicon and carbon that is extremely hard. Silicon carbide has been a crucial component of cutting tools, grinding wheels, and sandpaper since the late 19th century. In 1891, the American inventor Edward G. Acheson discovered silicon carbide while attempting to make an artificial diamond. Today, silicon carbide elements are used in the melting of glass and nonferrous metals, the heat treatment of metals, float glass manufacture, ceramics, and electronic component production, igniters   Table 3 of the edge partition.
The Z 2 (G)-index is determined as:  Table 3. The PM2(G)-index is determined as: Proof: The edges E are classified into 5 classes. By using Table 3 Proof: The HM-index is a vertex degree-related index that can be calculated by using Table 3 and the mathematical formula for this index.  Table 3.  The RRR(G)-index is computed as:  Table 3 and its definition, the F-index is easily computed as follows:  Numerical computational procedures are the approaches used to design mathematical problems that can be solved using arithmetic operations. The comparison of all outcomes can be easily observed from the data given in Table 4. These calculated values are used to draw the 3D graphs. The numerical values of all the topological indices are listed below in Table 4.

Graphical Analysis of Data
In this section, the results are presented and discussed using graphs. It is important to note that the estimated areas and graphs below show how t and h affect each topological index. These examples make it simpler to understand how the other topological indexes react differently to the parameters t and h. Graphs and charts summarized a lot of information in simple formats that express key ideas simply and effectively. According to the type of data, there are several graph types, including bar charts, line graphs, area graphs, scatter plots, pie charts, pictographs, column charts, and bubble charts. The 3D axis graphs are used here to understand the relationship between the T-indices and the physical-chemical properties of the silicon structures. The x-and y-axes are used to represent the input values of parameters t and h, but the 3D space is used to display the outcomes. The variants in all the indices are described in Fig. 3. M.K. Siddiqui submitted an article that describes many classical indices for silicon carbide 35 .  The behavior and variation of any TI can be observed by the numerical values shown in Table 5, but graphs are a better way to express the data. The smaller measurements are used to understand the changes as t = h = 1, 2, 3, 4, and 5. comparable to those of the previous indices and even more precise for some characteristics. So these indices also have a good correlation with the properties of silicon carbide. The values of TIs for t = h = 1, 2, 3, 4, and 5 are given in Table 6.  The graphical representation of the bi-distance first and second Zagreb indices is shown in Fig.  5.

Si2C3-I[t, h]
The significant connection between both indices is made clear by the comparison of both the classical and bi-distance indices. Bi-distant indices have excellent correlation coefficients that accurately represent a wide range of physical and chemical characteristics of various organic and inorganic materials.

Conclusion:
The graphic is a simple technique to portray the chemical nature of a database association. Graphs are an importan above, which analyzed fixed variables connected to chemical structure graphs. In chemistry, silicon carbide, Si2C3-I[t, h], plays a crucial role, especially in manufacturing techniques and host-guest collisions. Silicon carbide is often used in protective jackets, automobile clutches, vehicle brake pads, LED bulbs, and sensors. The effect of various multiplicative implementations of degree-based topological invariants of silicon carbide, Si2C3-I[t, h] was explored. The results of this study can be used to better understand the biological activities and physical characteristics of silicon carbide.
Similar to the classical degree-based approach, it is also a good way to estimate the properties of graphs because of the good correlation with the experimental features of Si2C3-I [t, h].
Although a huge number of indices and techniques are proposed to examine the geometry and characteristics of various chemical structures, these indices are still insufficient to analyze several features of chemical and non-chemical networks. As a result, topological indices will be increasingly important in the future.
A similar study might be conducted for many chemical substances that would be valuable to chemists in their future research. New methodologies, such as the bi-distance method, can be created and applied to all degree-or distancerelated topological indices. Topological indices can be used to investigate a wide range of complicated structures. Because classical degree-based indices are extremely strong and powerful, these indices will be applied to the structure that has to be explored. The most significant aspect of the significance of topological indices is the correlation coefficient values, which must be strong.

Data Availability
In this article, no data were utilized.

Funding Statement
This research received no funding.