# On Face Index of Silicon Carbides

Authors: Xiujun Zhang; Ali Raza; Asfand Fahad; Muhammad Kamran Jamil; Muhammad Anwar Chaudhry; Zahid Iqbal
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

Several graph invariants have been defined and studied, which present applications in nanochemistry, computer networks, and other areas of science. One vastly studied class of the graph invariants is the class of the topological indices, which helps in the studies of chemical, biological, and physical properties of a chemical structure. One recently introduced graph invariant is the face index, which can assist in predicting the energy and the boiling points of the certain chemical structures. In this paper, we drive the analytical closed formulas of face index of silicon carbidesSi2C3−Ia,b, Si2C3−IIa,b, Si2C3−IIIa,b, and SiC3−IIIa,b.


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## Body

## 1. 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. The 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 [1–5], studies on chemistry and drugs see [6–10], and other applications, see [11–13]. 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. The notions of a planar graph, its faces, and an infinite face are well known in the literature. LetG be a graph with vertex set VG, edge set EG, and face set FG. 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. The incidency of w to the face f is represented by w∼f. The degree of a face f in G is given as df=∑w∼fdw. 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. They 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(1)FIG=∑f∈FGdf=∑w∼f∈FGdw,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. These layers form tetrahedrally oriented molecules of C and Si atoms, with a very short bound length and hence a very high bound strength. This 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 [20–23]. There are several silicon carbides which we are going to study in this paper, such as Si2C3−Ia,b, Si2C3−IIa,b, Si2C3−IIIa,b, and SiC3−IIIa,b. Several papers have been devoted to the study of silicon, carbon-based structures, for details, see [24–26]. The 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 Section2 are consistent with this formula.Theorem 1.
For a finite, connected, and planar graphG with vertex set VG, edge set EG, and face set FG, we have(2)VG−EG+FG=2.


## 2. Results

In this section, we investigate the exact formulas of the face index forSi2C3−Ia,b, Si2C3−IIa,b, Si2C3−IIIa,b, and SiC3−IIIa,b. To find the face indices of the molecular graphs, we partitioned the face set depending on the degrees of each face.


### 2.1. Face Index forSi2C3−IIa,b and Graphical Representations

The molecular graphs of silicon carbidesSi2C3−Ia,b are given in Figure 1. Figure 1(a) consists of unit cell of the silicon carbide Si2C3−Ia,b and Si2C3−Ia,b for a=1 and b=1. Figure 1(b) is Si2C3−Ia,b for a=3 and b=4.Figure 1
The unit cell andSi2C3−I3,4, respectively.
(a)(b)Theorem 2.
LetG=Si2C3−Ia,b, where a,b≥1. Then,(3)FIG=70a−34,ifb=190ab−20a−30b−4,ifa,b≠1183b−1,ifa=1.Proof.
We consider the following three cases:Case-I Ifb=1, let fj and fj denote the face with the property ∑w∼fjdw=j and the number of such faces, respectively. The structure Si2C3−Ia,1 contains four types of internal faces f14, f16, f18, and f20 and an external face, f∞. So, the face index of Si2C3-Ia,1 is(4)FIG=∑w∼f∈FGdw=∑w∼f14dw+∑w∼f16dw+∑w∼f18dw+∑w∼f20dw+∑w∼f∞dw=f1414+f1616+f1818+f2020+20a=14a−1+16a−1+182+20a−2+20a=70a−34.Case-II Whena,b≠1, then, we notice that the structure Si2C3−Ia,b contains seven types of faces, which are f14, f15, f16, f18, f19, f20, and f21 and an external face, f∞. Moreover, the sum of vertex degrees of external face is 20a+30b−1. In each row of Table 1, the number of internal faces is written.
The face index ofSi2C3−Ia,b is(5)FIG=∑w∼f∈FGdw=∑w∼f14dw+∑w∼f15dw+∑w∼f16dw+∑w∼f18dw+∑w∼f19dw+∑w∼f20dw+∑w∼f21dw+∑w∼f∞dw=f1414+f1515+f1616+f1818+f1919+f2020+f2121+20a+30b−1=14a+2b−3+15b−12a−3+16a−1+18ab−a−b+3+192b−1+20a−2+21b−12a−3+20a+30b−1=90ab−20a−30b−4.Case-III Whena=1, the structure Si2C3−Ia,b contains three types of internal faces f13, f16, and f17 and an external face f∞ with sum of degrees 24b−4. Moreover, f13=b−1, f16=1, and f17=b−1. So, the face index of Si2C3−Ia,b is(6)FIG=∑w∼f∈FGdw=∑w∼f13dw+∑w∼f16dw+∑w∼f17dw+∑w∼f∞dw=f1313+f1616+f1717+24b−4=13b−1+161+17b−1+24b−4=183b−1,which completes the proof.Table 1
Numbers off14, f15, f16, f18, f19, f20, and f21 with given number of rows.
Rowsf14f15f16f18f19f20f212a + 12a−3a−1a + 12a−22a−33a + 32(2a−3)a−12a4a−22(2a−3)4a + 53(2a−3)a−13a−16a−23(2a−3).....———.....———.....———Ba + 2b−3(b−1) (2a−3)a−1ab−a−b + 32(b−1)a−2(b−1) (2a−3)Remark 1.
It is interesting to observe thatVSi2C3−Ia,b=10ab, ESi2C3−Ia,b=15ab−2a−3b, and FSi2C3−Ia,b=5ab−2a−3b+2. Hence, the structure Si2C3−Ia,b satisfies the Euler formula.
Now, we present the results obtained forSi2C3−Ia,b in a graphical way in 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. The 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.Figure 2
FISi2C3−Ia,1, FISi2C3−Ia,b, and FISi2C3−I1,b, respectively.
(a)(b)(c)


### 2.2. Face Index forSi2C3−IIa,b and Graphical Representations

The molecular graphs of silicon carbidesSi2C3−IIa,b are given in Figure 3, which consists of unit cell of silicon carbide Si2C3−IIa,b and Si2C3−IIa,b for a=4 and b=1.Figure 3
The unit cell andSi2C3−II4,1, respectively.
(a)(b)Theorem 3.
LetH=Si2C3−IIa,b, where a,b≥1. Then, the face index of H is given as(7)FIH=183a−2,ifb=1,90ab−30a+b−15,ifa,b≠1,183b−2,ifa=1.Proof.
We consider the following three cases.Case-I Ifb=1, then the graph of Si2C3−IIa,1 consists of two faces, an internal face f15 with f15=2a−2 and an external face, f∞ having the sum of vertex degrees 24a−6. The rest follows from the definition of the face index.Case-II Whena,b≠1, then, we notice that the structure Si2C3−IIa,b contains three types of internal faces, which are f16, f17, and f18 and an external face, f∞. Moreover, the sum of vertex degrees of external face is 30a+b−45. The number of internal faces in each row is given in Table 2.Table 2
Numbers off16, f17, and f18 with given number of rows.
No. of rowsf16f17f1822a2a3a−532a + 22a + 28a−1242a + 42a + 413a−19............B2(a + b)−42(a + b)−45ab−7(a + b) + 9The rest follows from the definition of the face index.Case-III Whena=1, the structure Si2C3−IIa,b contains an internal face f15 and an external face f∞ with sum of degrees 24b−6. Moreover, f15=2b−2. The rest follows from the definition of the face index.Remark 2.
It is interesting to observe thatVSi2C3−IIa,b=10ab, ESi2C3−IIa,b=15ab−2a−3b, and FSi2C3−IIa,b=5ab−2a−3b+2. Hence, the structure Si2C3−IIa,b satisfies the Euler formula.
With the same settings as in Section2.1, we present the results obtained for Si2C3−IIa,b in graphical way in Figure 4.Figure 4
FISi2C3−IIa,1, FISi2C3−IIa,b, and FISi2C3−II1,b, respectively.
(a)(b)(c)


### 2.3. Face Index forSiC3−IIIa,b and Graphical Representations

The molecular graphs of silicon carbidesSiC3−IIIa,b are given in Figures 5 and 6. Figure 5(a) is for a=5 and b=1, Figure 5(b) is for a=5 and b=2, and Figure 6 consists of unit cell and SiC3−IIIa,b for a=5 and b=5.Figure 5
(a)SiC3−III5,1 and (b) SiC3−III5,2.
(a)(b)Figure 6
The unit cell andSiC3−III5,5, respectively.
(a)(b)Theorem 4.
LetK=SiC3−IIIa,b, where a,b≥1. Then,(8)FIK=32a−18;forb=172ab−30a−20b−7;forb>1.Proof.
Whenb=1, then the graph of SiC3−IIIa,1 consists of two faces, an internal face f14 with f14=a−1, and an external face, f∞ having the sum of degrees 18a−4. The rest follows from the definition of the face index. Now, when b>1, then we notice that the structure SiC3−IIIa,b contains seven types of internal faces which are f13, f15, f17, f18, f20, f22, and f24 and an external face, f∞. Moreover, the sum of vertex degrees of external face is 30a+17b−30. In each row of Table 3, the number of internal faces is written.Table 3
Numbers off13, f15, f17, f18, f20, f22, and f24 with given number of rows.
Rowsf13f15f17f18f20f22f24213(a−1)a−1—1a−1—325(a−1)a−1a−11Aa−1437(a−1)a−12(a−1)1a + 12(a−1)........................Bb−1(2b−1) (a−1)a−1(b−2) (a−1)1a + b−3(b−2) (a−1)The rest follows from the definition of the face index.Remark 3.
It is interesting to observe thatVSiC3−IIIa,b=8ab, ESiC3−IIIa,b=12ab−3a−2b, and FSiC3−IIIa,b=4ab−3a−2b+2. Hence, the structure SiC3−IIIa,b satisfies the Euler formula.
With the same settings as in Section2.1, we present the results obtained for SiC3−IIIa,b in a graphical way in Figure 7.Figure 7
FISiC3−IIIa,1, FISiC3−IIIa,b, and FISiC3−III1,b, respectively.
(a)(b)(c)


### 2.4. Face Index forSiC3−IIIa,b and Graphical Representations

The molecular graphs of silicon carbidesSiC3−IIIa,b are given in Figure 8.Figure 8
The unit cell andSiC3−III5,4, respectively.
(a)(b)Theorem 5.
LetL=SiC3−IIIa,b, where a,b≥1. Then,(9)FIL=70a−32,forb=1,90ab−20a−30b−2,forb>1.Proof.
Whenb=1, then the graph of Si2C3−IIIa,1 consists of four types of faces, three internal faces f14, f20, and f22 and an external face f∞ having the sum of degrees 20a. Moreover, the values of f14, f20, and f22 are 2a−1, 2, and a−2, respectively. The rest follows from the definition of the face index. Now, when b>1, then we notice that the structure SiC3−IIIa,b contains eight types of internal faces, which are f14, f15, f17, f18, f21, f22,f23, and f24 and an external face, f∞. Moreover, the sum of vertex degrees of external face is 20a+24b−1. In each row of Table 4, the number of internal faces is written.Table 4
Numbers off14, f15, f17, f18, f21, f22, f23, and f24 with given number of rows.
No. of rowsf14f15f17f18f21f22f23f2422(a−1)2(a−1)2(2a−3)4—2(a−2)—32(a−1)4(a−1)42(2a−3)422(a−2)a−242(a−1)6(a−1)63(2a−3)442(a−2)2(a−2)........................B2(a−1)2(b−1) (a−1)2(b−1)(b−1) (2a−3)42(b−2)2(a−2)(b−2) (a−2)The rest follows from the definition of the face index.Remark 4.
It is interesting to observe thatVSi2C3−IIIa,b=10ab, ESi2C3−IIIa,b=15ab−2a−3b, and FSi2C3−IIIa,b=5ab−32−3b+2. Hence, the structure Si2C3−IIIa,b satisfies the Euler formula.
With the same settings as in Section2.1, we present the results obtained for Si2C3−IIIa,b in a graphical way in Figure 9.Figure 9
FISi2C3−IIIa,1, FISi2C3−IIIa,b, and FISi2C3−III1,b, respectively.
(a)(b)(c)


## 2.1. Face Index forSi2C3−IIa,b and Graphical Representations

The molecular graphs of silicon carbidesSi2C3−Ia,b are given in Figure 1. Figure 1(a) consists of unit cell of the silicon carbide Si2C3−Ia,b and Si2C3−Ia,b for a=1 and b=1. Figure 1(b) is Si2C3−Ia,b for a=3 and b=4.Figure 1
The unit cell andSi2C3−I3,4, respectively.
(a)(b)Theorem 2.
LetG=Si2C3−Ia,b, where a,b≥1. Then,(3)FIG=70a−34,ifb=190ab−20a−30b−4,ifa,b≠1183b−1,ifa=1.Proof.
We consider the following three cases:Case-I Ifb=1, let fj and fj denote the face with the property ∑w∼fjdw=j and the number of such faces, respectively. The structure Si2C3−Ia,1 contains four types of internal faces f14, f16, f18, and f20 and an external face, f∞. So, the face index of Si2C3-Ia,1 is(4)FIG=∑w∼f∈FGdw=∑w∼f14dw+∑w∼f16dw+∑w∼f18dw+∑w∼f20dw+∑w∼f∞dw=f1414+f1616+f1818+f2020+20a=14a−1+16a−1+182+20a−2+20a=70a−34.Case-II Whena,b≠1, then, we notice that the structure Si2C3−Ia,b contains seven types of faces, which are f14, f15, f16, f18, f19, f20, and f21 and an external face, f∞. Moreover, the sum of vertex degrees of external face is 20a+30b−1. In each row of Table 1, the number of internal faces is written.
The face index ofSi2C3−Ia,b is(5)FIG=∑w∼f∈FGdw=∑w∼f14dw+∑w∼f15dw+∑w∼f16dw+∑w∼f18dw+∑w∼f19dw+∑w∼f20dw+∑w∼f21dw+∑w∼f∞dw=f1414+f1515+f1616+f1818+f1919+f2020+f2121+20a+30b−1=14a+2b−3+15b−12a−3+16a−1+18ab−a−b+3+192b−1+20a−2+21b−12a−3+20a+30b−1=90ab−20a−30b−4.Case-III Whena=1, the structure Si2C3−Ia,b contains three types of internal faces f13, f16, and f17 and an external face f∞ with sum of degrees 24b−4. Moreover, f13=b−1, f16=1, and f17=b−1. So, the face index of Si2C3−Ia,b is(6)FIG=∑w∼f∈FGdw=∑w∼f13dw+∑w∼f16dw+∑w∼f17dw+∑w∼f∞dw=f1313+f1616+f1717+24b−4=13b−1+161+17b−1+24b−4=183b−1,which completes the proof.Table 1
Numbers off14, f15, f16, f18, f19, f20, and f21 with given number of rows.
Rowsf14f15f16f18f19f20f212a + 12a−3a−1a + 12a−22a−33a + 32(2a−3)a−12a4a−22(2a−3)4a + 53(2a−3)a−13a−16a−23(2a−3).....———.....———.....———Ba + 2b−3(b−1) (2a−3)a−1ab−a−b + 32(b−1)a−2(b−1) (2a−3)Remark 1.
It is interesting to observe thatVSi2C3−Ia,b=10ab, ESi2C3−Ia,b=15ab−2a−3b, and FSi2C3−Ia,b=5ab−2a−3b+2. Hence, the structure Si2C3−Ia,b satisfies the Euler formula.
Now, we present the results obtained forSi2C3−Ia,b in a graphical way in 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. The 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.Figure 2
FISi2C3−Ia,1, FISi2C3−Ia,b, and FISi2C3−I1,b, respectively.
(a)(b)(c)


## 2.2. Face Index forSi2C3−IIa,b and Graphical Representations

The molecular graphs of silicon carbidesSi2C3−IIa,b are given in Figure 3, which consists of unit cell of silicon carbide Si2C3−IIa,b and Si2C3−IIa,b for a=4 and b=1.Figure 3
The unit cell andSi2C3−II4,1, respectively.
(a)(b)Theorem 3.
LetH=Si2C3−IIa,b, where a,b≥1. Then, the face index of H is given as(7)FIH=183a−2,ifb=1,90ab−30a+b−15,ifa,b≠1,183b−2,ifa=1.Proof.
We consider the following three cases.Case-I Ifb=1, then the graph of Si2C3−IIa,1 consists of two faces, an internal face f15 with f15=2a−2 and an external face, f∞ having the sum of vertex degrees 24a−6. The rest follows from the definition of the face index.Case-II Whena,b≠1, then, we notice that the structure Si2C3−IIa,b contains three types of internal faces, which are f16, f17, and f18 and an external face, f∞. Moreover, the sum of vertex degrees of external face is 30a+b−45. The number of internal faces in each row is given in Table 2.Table 2
Numbers off16, f17, and f18 with given number of rows.
No. of rowsf16f17f1822a2a3a−532a + 22a + 28a−1242a + 42a + 413a−19............B2(a + b)−42(a + b)−45ab−7(a + b) + 9The rest follows from the definition of the face index.Case-III Whena=1, the structure Si2C3−IIa,b contains an internal face f15 and an external face f∞ with sum of degrees 24b−6. Moreover, f15=2b−2. The rest follows from the definition of the face index.Remark 2.
It is interesting to observe thatVSi2C3−IIa,b=10ab, ESi2C3−IIa,b=15ab−2a−3b, and FSi2C3−IIa,b=5ab−2a−3b+2. Hence, the structure Si2C3−IIa,b satisfies the Euler formula.
With the same settings as in Section2.1, we present the results obtained for Si2C3−IIa,b in graphical way in Figure 4.Figure 4
FISi2C3−IIa,1, FISi2C3−IIa,b, and FISi2C3−II1,b, respectively.
(a)(b)(c)


## 2.3. Face Index forSiC3−IIIa,b and Graphical Representations

The molecular graphs of silicon carbidesSiC3−IIIa,b are given in Figures 5 and 6. Figure 5(a) is for a=5 and b=1, Figure 5(b) is for a=5 and b=2, and Figure 6 consists of unit cell and SiC3−IIIa,b for a=5 and b=5.Figure 5
(a)SiC3−III5,1 and (b) SiC3−III5,2.
(a)(b)Figure 6
The unit cell andSiC3−III5,5, respectively.
(a)(b)Theorem 4.
LetK=SiC3−IIIa,b, where a,b≥1. Then,(8)FIK=32a−18;forb=172ab−30a−20b−7;forb>1.Proof.
Whenb=1, then the graph of SiC3−IIIa,1 consists of two faces, an internal face f14 with f14=a−1, and an external face, f∞ having the sum of degrees 18a−4. The rest follows from the definition of the face index. Now, when b>1, then we notice that the structure SiC3−IIIa,b contains seven types of internal faces which are f13, f15, f17, f18, f20, f22, and f24 and an external face, f∞. Moreover, the sum of vertex degrees of external face is 30a+17b−30. In each row of Table 3, the number of internal faces is written.Table 3
Numbers off13, f15, f17, f18, f20, f22, and f24 with given number of rows.
Rowsf13f15f17f18f20f22f24213(a−1)a−1—1a−1—325(a−1)a−1a−11Aa−1437(a−1)a−12(a−1)1a + 12(a−1)........................Bb−1(2b−1) (a−1)a−1(b−2) (a−1)1a + b−3(b−2) (a−1)The rest follows from the definition of the face index.Remark 3.
It is interesting to observe thatVSiC3−IIIa,b=8ab, ESiC3−IIIa,b=12ab−3a−2b, and FSiC3−IIIa,b=4ab−3a−2b+2. Hence, the structure SiC3−IIIa,b satisfies the Euler formula.
With the same settings as in Section2.1, we present the results obtained for SiC3−IIIa,b in a graphical way in Figure 7.Figure 7
FISiC3−IIIa,1, FISiC3−IIIa,b, and FISiC3−III1,b, respectively.
(a)(b)(c)


## 2.4. Face Index forSiC3−IIIa,b and Graphical Representations

The molecular graphs of silicon carbidesSiC3−IIIa,b are given in Figure 8.Figure 8
The unit cell andSiC3−III5,4, respectively.
(a)(b)Theorem 5.
LetL=SiC3−IIIa,b, where a,b≥1. Then,(9)FIL=70a−32,forb=1,90ab−20a−30b−2,forb>1.Proof.
Whenb=1, then the graph of Si2C3−IIIa,1 consists of four types of faces, three internal faces f14, f20, and f22 and an external face f∞ having the sum of degrees 20a. Moreover, the values of f14, f20, and f22 are 2a−1, 2, and a−2, respectively. The rest follows from the definition of the face index. Now, when b>1, then we notice that the structure SiC3−IIIa,b contains eight types of internal faces, which are f14, f15, f17, f18, f21, f22,f23, and f24 and an external face, f∞. Moreover, the sum of vertex degrees of external face is 20a+24b−1. In each row of Table 4, the number of internal faces is written.Table 4
Numbers off14, f15, f17, f18, f21, f22, f23, and f24 with given number of rows.
No. of rowsf14f15f17f18f21f22f23f2422(a−1)2(a−1)2(2a−3)4—2(a−2)—32(a−1)4(a−1)42(2a−3)422(a−2)a−242(a−1)6(a−1)63(2a−3)442(a−2)2(a−2)........................B2(a−1)2(b−1) (a−1)2(b−1)(b−1) (2a−3)42(b−2)2(a−2)(b−2) (a−2)The rest follows from the definition of the face index.Remark 4.
It is interesting to observe thatVSi2C3−IIIa,b=10ab, ESi2C3−IIIa,b=15ab−2a−3b, and FSi2C3−IIIa,b=5ab−32−3b+2. Hence, the structure Si2C3−IIIa,b satisfies the Euler formula.
With the same settings as in Section2.1, we present the results obtained for Si2C3−IIIa,b in a graphical way in Figure 9.Figure 9
FISi2C3−IIIa,1, FISi2C3−IIIa,b, and FISi2C3−III1,b, respectively.
(a)(b)(c)


## 3. 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. These 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.



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*Source: 6048438-2020-08-01.xml*