Abstract

The silicon material has provoked and stimulated significant research concern to a considerable extent taking into account its marvelous mechanical, optical, and electronic properties. Naturally, silicons are semiconductors and are utilized in the formation of various materials. For example, it is used in assembling the electronic based gadgets. In this article, we have studied the structure of silicon carbide and and then continued to discuss some degree grounded topological descriptors in association with their corresponding entropy measures. We extend this computation to the quantitative and pictorial comparisons which could be beneficial in the structure amendment for effective implementation.

1. Introduction

Chemical graph theory is a fascinating branch of mathematics and is the combination of chemistry and graph theory. In molecular graph, molecules are designed mathematically. In a molecular graph, the vertices and edges portrayed the atoms and the bonds correspondingly. Various approaches are employed on molecular graphs to acquire various structure properties of corresponding compounds. A numerical worth treated theoretically from the atomic graph is named as topological index. It is connected with synthetic constitution exhibiting for association of substance structure with miscellaneous physical and natural exercises and concoction properties. The degree-based topological indices collaborate actively in the exploitation of different fields specifically in pharmaceutical and commercial chemistry. In [1], Randić index is originated, while index, index, first Zagreb index, and second Zagreb were discussed in [2, 3]. In [4], hyper-Zagreb index was introduced. In [5], forgotten index was initiated. Furtula [6] established the concept of index. Balaban [7, 8] laid the foundation of Balaban index. The redefined first, second, and third Zagreb indices were investigated in [9]. The forth atom bond connectivity index was presented in [10]. The fifth geometric arithmetic index is declared in [11]. The specific topological indices of some graphs are discussed in [12, 13]. An ordered pair , composed of a nonempty set of vertices and as a set of edges, is called a graph. If any two vertices and are attached with each other, they make an edge . The number of edges adjacent with the vertex is characterized as .

Entropy has been a comprehensive and transcendental approach in diverse areas of expertise diverging from logic and biology to physics and engineering. Entropy connects the conception of randomness and uncertainty with physical conditions which are constructed as transformation channels of information. This study represents the special case when these channels are the detectors investigating the response of a system. Formally, entropy came into consideration after the invention of the heat engine, as a consequence of pioneering investigation towards explaining thermodynamical techniques and expanding the efficacy of such machines [14, 15]. This investigation reveals that the entropy of a system or machine cannot decrease over time. In 1948, Claude Shannon [16] introduces a measure of uncertainty as entropy. It can be explicated as the rate of production of new information manufactured by the system [1719]. As stated by Shannon, uncertainty and information are two sides of the coin: a reduction in uncertainty is the same as the reception of a certain amount of information. Therefore, as the value of the entropy about a system is greater, the uncertainty about its response is also increased. In literature, many graph entropies are estimated; see [2023].

In 2014, Chen et al. [24] introduced the definition of the entropy of edge weighted graph . The entropy formula is represented in

2. Crystallographic Structure of

The most consistent framework of silicon carbon unicellular compound is anticipated which is grounded on the particle swarm optimization approach. It is rich in carbon. In the last decades, silicon carbon was considered to be the hardest material across the world. Due to its Mohs hardness rating, it resembles diamond. Silicon is of low cost and is nontoxic semiconductor. Numerous studies have been done in the refinement, magnification, and device assembling [25]. It is employed for all advanced electronic gadgets. The silicon carbide framework may seem like the honeycomb framework of graphene. Despite that, large scale search proposes that the framework of this compound is absolutely distinct compared to that of graphene. For silicon carbide layer, the minimum energy of - demonstrates a planar framework comprised of polygonal rings, where two pentagonal and four heptagonal rings encircling each hexagonal ring. See Figure 1 [26]. Every hexagonal ring carries out three and three atoms in which and atoms are positioned by turn on the vertices. Pentagonal rings are of two kinds in which one is comprised of two and three atoms. Also, three and four atoms establish the heptagonal rings. It is to be noted that there are no bonds in - sheet. Here denotes the total associated unit cells in each row while shows the total sum of associated rows with number of cells.

Figure 2 illustrates the way in which cells are associated in a row and the way of association of rows with each other. Also, and .

2.1. Formation of Formulas

Here, we join one unit cell with another unit cell and then continue this process horizontally till unit cells. Similar technique will be applied for vertical direction. As a result, we will obtain the sheet; see Figure 1. The vertex and edge partition are depicted in Tables 1 and 2, respectively.

Table 3 shows the edge partition of the chemical graph based on degree sum of end vertices.

2.2. Computation of Entropies for Crystallographic Structure of

This portion deals with the computation of topological indices and their corresponding graph entropies for the crystallographic structure of .

2.2.1. The Randić Index and Randić Entropy for

For , the Randić index and Randić entropy by using Table 2 are

For , by using Table 2, the Randić index and Randić entropy are

For , by using Table 2, the Randić index and Randić entropy are

For , by using and Table 2, the Randić index and Randić entropy are

2.2.2. The Index and Entropy of

By using Table 2, the index and entropy are

2.2.3. The Index and Entropy of

By using Table 2, the index and entropy are

2.2.4. The First Zagreb Index and First Zagreb Entropy of

By using Table 2, the first Zagreb index and entropy are

2.2.5. The Second Zagreb Index and Second Zagreb Entropy of

By using Table 2, the second Zagreb index and second Zagreb entropy are

2.2.6. The Hyper-Zagreb Index and Hyper-Zagreb Entropy of

By using Table 2, the hyper-Zagreb index and hyper-Zagreb entropy are

2.2.7. The Forgotten Index and Forgotten Entropy for

By using Table 2, the forgotten index and forgotten entropy are

2.2.8. The Augmented Zagreb Index and Augmented Zagreb Entropy of

By using Table 2, the index and entropy are

2.2.9. The Balaban Index and Balaban Entropy for

By using Table 2, the Balaban index and Balaban entropy are

2.2.10. The Redefined First Zagreb Index and Redefined First Zagreb Entropy for

By using Table 2, the redefined first Zagreb index and redefined first Zagreb entropy are and

2.2.11. The Redefined Second Zagreb Index and Redefined Second Zagreb Entropy for

By using Table 2, the redefined second Zagreb index and redefined second Zagreb entropy are and

2.2.12. The Redefined Third Zagreb Index and Redefined Third Zagreb Entropy for

By using Table 2, the redefined third Zagreb index and redefined third Zagreb entropy are and

2.2.13. The Fourth ABC Index and Fourth ABC Entropy of

By using Table 3, the fourth ABC index and fourth ABC entropy are

2.2.14. The Fifth Index and Fifth Entropy of

By using Table 3, the fifth index and fifth entropy are

3. Crystallographic Structure of

The framework of sheet contains only hexagonal rings. This framework is greater in energy than that of sheet. On the other hand, the composition of is quite different from sheet as bonds can be viewed and the ratio of atoms making dimers is half. Also, is more unstable than . The anticipated minimal energy frameworks and sheets have exceptional semiconducting axioms that could be employed for monoelectronic utilization. The mechanic strength of the silicon carbide sheets is also significant. It is widely known that graphene carries magnificent elastic properties having immense elastic constants. Former analyses have demonstrated that silicon carbide also holds well elastic axioms.

In Figure 3 [27, 28], denotes the total associated unit cells in each row while shows the total sum of associated rows with number of cells. Figure 4 illustrates the way in which cells are associated in a row and the way of association of rows with each other. Also, and

3.1. Methodology of Silicon Carbide Formulas

For silicon carbide , we join one unit cell with another unit cell and then continue this process horizontally till unit cells. Similar technique will be applied for vertical direction. As a result, we will obtain the sheet; see Figure 3. Moreover, Tables 4 and 5 are used for computation of vertices and edges, respectively.

The edge partition of established on the addition of degree of terminal vertices of every edge is illustrated in Table 6.

3.2. Computation of Entropies for Crystallographic Structure of

This portion deals with the computation of topological indices and their corresponding graph entropies for the crystallographic structure of .

3.2.1. The Randić Index and Randić Entropy for

For , the Randić index and Randić entropy by Table 5 are

For , by using Table 5, the Randić index and Randić entropy are and

For , by using Table 2, the Randić index and Randić entropy are and