On Face Index of Silicon Carbides

School of Information Science and Engineering, Chengdu University, Chengdu 610106, China Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61110, Pakistan Department ofMathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University, 14 Ali Road, Lahore, Pakistan Department of Mathematics and Statistics, Institute of Southern Punjab, Multan 66000, Pakistan School of Natural Sciences, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan


Introduction
Several invariants assigning a matrix, a polynomial, a numeric number, or a sequence (of numbers) to a graph have been defined and studied during the last few decades. One such significant class among such invariants is the class of topological indices (TIs), which assigns a number to the given graph. e value of a TI of a molecular structure is dependent on its shape, size, symmetry, patterns of the bonds, and the contents of heteroatoms in it. Consequently, the notion of the TI provides the quantitative characterization of the molecular structures. Several researchers have studied different aspects and applications of the TIs. For contents related to studies on lower/upper bounds maxima/ minima of TIs, see [1][2][3][4][5], studies on chemistry and drugs see [6][7][8][9][10], and other applications, see [11][12][13]. Another vital contribution of the study of the TIs is their effectiveness in studying the different aspects of new drugs and chemical compounds, which is an immense need of the medical science and industry. For details, we refer [14,15] to the readers. Consequently, computing and studying the behavior of the values of the TIs of the molecular structures provide significant information and hence is one of the trends in modern research.
Before proceeding further with the study of a specific index, we set the notations used in this paper. e notions of a planar graph, its faces, and an infinite face are well known in the literature. Let G be a graph with vertex set V(G), edge set E(G), and face set F(G). A face f is said to be incident with an edge e, whenever e is among the edges which surrounds f. Moreover, face is said to be incident with a vertex w, whenever w is incident to an edge e which surrounds f. e incidency of w to the face f is represented by w ∼ f. e degree of a face f in G is given as d(f) � w∼f d w . For the notions and notations not given here, we refer [16] to the readers. Recently, Jamil et al. [17] introduced a novel topological index named as the face index. ey showed that the face index can help to predict the boiling points and the energy of selected benzenoid hydrocarbons with the correlation coefficient r > 0.99. For a planar graph G, the face index (FI) can be defined as where w ∼ f represents the incidency of the vertex w with the face f. On the contrary, the Silicon Carbide (SiC) is the first and foremost material that contained the covalent bound of C and Si atoms, typically in biatomic layers. ese layers form tetrahedrally oriented molecules of C and Si atoms, with a very short bound length and hence a very high bound strength. is is the base of extremely high mechanically and chemical stability of SiC [18,19]. SiC occurs in nature as the incredibly uncommon mineral moissanite. SiC being one of the most extensively used wide bandgap materials, performs a vital role in power industries by setting new principles in power savings as rectifiers or switches in the system for data centers, solar cells, wind turbines, and electric vehicles, as well as high temperature and radiation tolerant electronic applications [20][21][22][23].
ere are several silicon carbides which we are going to study in this paper, such as . Several papers have been devoted to the study of silicon, carbon-based structures, for details, see [24][25][26].
e main objective of this paper is to find the analytic formula of the face index of these silicon carbides. Moreover, we also present the graphical analysis of the obtained results. For this, we use the Chemsketch for plotting the figures of silicon carbides, Maple for calculations, and MATLAB for graphical analysis.
Before presenting the results, we include Euler's formula for planar graphs. Evidently, we may observe that the silicon structures presented in Section 2 are consistent with this formula. (2)

Results
In this section, we investigate the exact formulas of the face index for To find the face indices of the molecular graphs, we partitioned the face set depending on the degrees of each face.

Face Index for Si 2 C 3 − II[a, b] and Graphical
Representations. e molecular graphs of silicon carbides Si 2 C 3 − I[a, b] are given in Figure 1.  Figure 1 Proof. We consider the following three cases: 2 Discrete Dynamics in Nature and Society Case-II When a, b ≠ 1, then, we notice that the structure  Rows   Discrete Dynamics in Nature and Society 3 Case-III When a � 1, the structure which completes the proof.  Figure 2. Every silicon carbide structure depends on two variables a and b. We use two kinds of graphs to show the outcomes of face index. One is the 2D graph where the reliance of a face index is drawn against one variable of the structure a or b while other is kept fixed. Here, we demonstrate 2D graphs for b � 1 and change a, and for a � 1 and change b. e second tool is the 3D graph where reliance of a face index is drawn against both parameters, and we gained a surface that tells about the trends of face index against both parameters a and b at the same time.

Face Index for Si 2 C 3 − II[a, b] and Graphical
Representations.
e molecular graphs of silicon carbides Proof. We consider the following three cases.

Remark 2. It is interesting to observe that
With the same settings as in Section 2.1, we present the results obtained for Si 2 C 3 − II[a, b] in graphical way in Figure 4.

Face Index for SiC 3 − III[a, b] and Graphical
Representations.
e molecular graphs of silicon carbides SiC 3 − III[a, b] are given in Figures 5 and 6. Figure 5(a) is for a � 5 and b � 1, Figure 5 III[a, b], where a, b ≥ 1. en,    With the same settings as in Section 2.1, we present the results obtained for SiC 3 − III[a, b] in a graphical way in Figure 7.

Face Index for SiC 3 − III[a, b] and Graphical
Representations.
e molecular graphs of silicon carbides SiC 3 − III[a, b] are given in Figure 8.   Table 4, the number of internal faces is written. □ Remark 4. It is interesting to observe that Hence, the structure Si 2 C 3 − III[a, b] satisfies the Euler formula.
With the same settings as in Section 2.1, we present the results obtained for Si 2 C 3 − III[a, b] in a graphical way in Figure 9.

Conclusions
An ongoing direction in mathematical and computational chemistry is the assessment of various properties of molecular structures with the help of numerical graph descriptors. ese invariants have also established the feasible applications in QSAR/QSPR studies, which are beneficial for the new molecular designs, drug discoveries, and hazard estimation of chemicals. Hence, the novel index, named as the face index for the molecular graphs of silicon carbides, is presented through the numerical way, and we have also displayed our numeric outcomes in the graphical way. Our consequences could be applicable in assessing and comparing the properties of these molecular structures.

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

Conflicts of Interest
e authors declare no conflicts of interest.   Discrete Dynamics in Nature and Society 7