Valency-Based Descriptors for Silicon Carbides, Bismuth(III) Iodide, and Dendrimers in Drug Applications

School of Pharmacy, Anhui Xinhua University, Hefei 230088, China Department of Mathematics, University of Management and Technology, Lahore 54000, Pakistan Department of Mathematics, COMSATS University of Islamabad, Lahore Campus, Lahore 54000, Pakistan Department for Management of Science and Technology Development, Ton Duc ,ang University, Ho Chi Minh City, Vietnam Faculty of Applied Sciences, Ton Duc ,ang University, Ho Chi Minh City, Vietnam


Introduction
Mathematical chemistry provides tools such as polynomials and functions that depend upon the information hidden in the symmetry of graphs of chemical compounds and helps to predict properties of the understudy molecular compound without the use of quantum mechanics. A topological index is a numerical parameter of a graph and depicts its topology. It describes the structure of molecules numerically and are used in the development of qualitative structure activity relationships (QSARs). ere are three kinds of topological indices, namely, degree-based, distance-based, and surfacebased topological indices. Lot of research has been done on degree-based topological indices, for example, see [1][2][3][4][5][6][7][8][9]. Degree-based topological indices correlate the structure of the molecular compound with its various physical properties, biological activities, and chemical reactivity [10][11][12][13][14].
Boiling point, heat of formation, fracture toughness, strain energy, and rigidity of a molecule are strongly connected to its graphical structure. e first topological index was introduced by Wiener when he was studying the boiling point of alkanes [15], which is now known as the Wiener index [16][17][18][19][20]. In 1975, Milan Randić introduced a simple topological index called the Randić index [21]. Many research papers and survey papers have been written on this graph invariant due to its interesting mathematical properties and valuable applications in chemistry [22][23][24][25][26][27]. e other oldest topological indices are Zagreb indices defined by Gutman and Trinajstic in [28] and are one of the most studied topological indices [29][30][31][32][33]. Topological indices are helpful in guessing properties of concerned compounds and are used in QSPRs [34][35][36][37]. ere are more than 148 topological indices in the literature [38][39][40][41][42], but none of them are able to guess all the properties of the concerned compound (together they do it to some extent). erefore, there is always room to define new topological indices [43]. Recently, in 2017, the first and second Gourava indices [44] were defined as (1) In the same year, the first and second hyper-Gourava indices [45] have been defined as Note that GO 1 (G) � M 1 (G) + M 2 (G), GO 2 (G) � M 1 (G)M 2 (G), HGO 1 (G) � H 1 (G) + H 2 (G) + 2M 1 (G) + M 2 (G), and HGO 2 (G) � H 1 (G)H 2 (G). In this paper, the aim is to compute Gourava indices and hyper-Gourava indices for silicone carbides, bismuth triiodide, and dendrimers and their graphical representations.

Methodology
To compute our results, first we constructed the graph of the concerned molecular compounds and counted the total number of vertices and edges. Secondly, we divided the edge set of concerned graphs into different classes based on the degrees of end vertices. By applying definitions of Gourava indices, we computed our desired results. We plotted our computed results by using Maple 2015 to see their dependencies on the involved parameters.

Gourava Indices
In this section, we present our main computational results.

Gourava Indices for Silicon Carbides.
Silicon carbide (SiC), also called carborundum, is a semiconductor containing silicon and carbon. It occurs in nature as the incredibly uncommon mineral Moissanite. Manufactured SiC powder has been created in mass since 1893 for use as an abrasive. Grains of silicon carbide are reinforced together by sintering to shape extremely hard ceramic production that are generally utilized in applications requiring high continuance, for example, vehicle brakes, vehicle clutches, and ceramic plates in impenetrable vests. Electronic utilizations of silicon carbide, for example, light-emitting diodes (LEDs) and locators in early radios, were first exhibited around 1907. SiC is utilized in semiconductor electronic devices that work at high temperatures or high voltages, or both. Huge single crystals of silicon carbide can be developed by the Lely technique, and they can be cut into gems known as manufactured Moissanite. SiC with a high surface zone can be created from SiO 2 contained in the plant material. Due to huge amount of application, silicone carbides have been studied extensively [6,42]. In this section, we computed Gourava indices for silicon carbides Si 2 C 3 − I[p, q],

Gourava Indices for Silicon Carbide Si
e molecular graphs of silicon carbide Si 2 C 3 − I[p, q] are shown in Figures 1-4, where Figure 1 shows the unit cell of silicone carbide, Figure 2 shows Si 2 C 3 I[p, q] for p � 4, q � 3, Figure 3 shows Si 2 C 3 I[p, q] for p � 4, q � 1, and Figure 4 shows Si 2 C 3 I[p, q] for p � 4, q � 3. e edge partition of the edge set of Si 2 C 3 − I[p, q] based on the degree of the end vertex is given in Table 1. q]. en, the first and second Gourava indices are Proof. From the edge partition of Si 2 C 3 I[p, q] given in Table 1, we have (1) e first Gourava index for Si 2 C 3 I[p, q] is (2) e second Gourava index for Si 2 C 3 I[p, q] is Proof. From the edge partition of Si 2 C 3 I[p, q] given in Table 1, we have (1) e first hyper-Gourava index for Si 2 C 3 I[p, q] is (2) e second hyper-Gourava index for Si 2 C 3 I[p, q] is e molecular graphs of silicon carbide Si 2 C 3 − II[p, q] are shown in Figures 5-8, where Figure 5 shows the unit cell of Si 2 C 3 − II[p, q], Figure 6 shows Si 2 C 3 − II[p, q] for p � 3, q � 3, Figure 7 shows Si 2 C 3 − II[p, q] for p � 5, q � 1, and Figure 8 shows Si 2 C 3 − II[p, q] for p � 5, q � 2. e edge partition of the edge set of Si 2 C 3 − II[p, q] based on the degree of the end vertex is given in Table 2.
Proof. From the edge partition of Si 2 C 3 − II[p, q] given in Table 2, we have (1) e first Gourava index for Si 2 C 3 I[p, q] is . en, the first and second hyper-Gourava indices are Proof. From the edge partition of Si 2 C 3 − II[p, q] given in Table 2, we have (2) e second hyper-Gourava index for □

Gourava Indices for Silicon Carbide Si
e unit cell of Si 2 C 3 III[p, q] is shown in Figure 9. e 2D lattice graphs of Si 2 C 3 − I[5, 1], Si 2 C 3 − I [5,2], and Si 2 C 3 − I [5,4] are shown in Figures 10-12, respectively. e edge partition of the edge set of Si 2 C 3 III[p, q] based on the degrees of end vertices is given in Table 3.

Theorem 5. Let G be the graph of silicon carbide
. en, the first and second Gourava indices are Proof. From the edge partition of Si 2 C 3 III[p, q] given in Table 3, we have Journal of Chemistry □ Theorem 6. Let G be the graph of silicon carbide en, the first and second hyper-Gourava indices are Proof. From the edge partition of Si 2 C 3 III[p, q] given in Table 3, we have (1) e first hyper-Gourava index for Si 2 (2) e second hyper-Gourava index for Si 2 □ Figure 13. e 2D lattice graphs of SiC 3 − III [5,1], SiC 3 − III [5,2], and SiC 3 − III [5,4] are shown in Figures 14-16, respectively. e edge partition of the edge set of SiC 3 − III[p, q] based on the degrees of end vertices is given in Table 4.

Gourava Indices for Silicon Carbide SiC
Proof. From the edge partition of the edge set of SiC 3 III[p, q] given in Table 4, we have (2) e second Gourava index for SiC 3 III[p, q] is Journal of Chemistry 5 Proof. From the edge partition of the edge set of SiC 3 III[p, q] given in (2) e second hyper-Gourava index for SiC 3 III[p, q] is In Figures 17-20, we can observe that the behavior of all indices is exponentially increasing with respect to the involved parameters. Codes for plotting the first and second Gourava indices for silicon carbide Si 2 C 3 I[p, r] are given as follows: 3.2. Gourava Indices for Bismuth Triiodide. BiI 3 is an inorganic compound which is the result of the reaction between iodine and bismuth, which inspired the enthusiasm for subjective inorganic investigations [46]. BiI 3 is an excellent inorganic compound and is very useful in qualitative inorganic analysis [47]. It has been proven that Bi-doped glass optical strands are one of the most promising dynamic laser media. Different kinds of Bi-doped fiber strands have been created and have been used to construct Bi-doped fiber lasers and optical loudspeakers [48]. Layered BiI 3 gemstones are considered to be a three-layered stack structure in which a plane of bismuth atoms is sandwiched between iodide 12pq − 12p − 8q + 8 particle planes to form a continuous plane [49]. e periodic superposition of the three layers forms diamond-shaped BiI 3 crystals with R − 3 symmetry [50,51]. A progressive stack of I − Bi − I layers forms a hexagonal structure with symmetry [52]. A jewel of BiI 3 has been integrated in [46]. We referred to [6] for the topological study of bismuth triiodide. 3 . e molecular graph of the unit cell of m − BiI 3 is shown in Figure 21. From Figure 22, we can see that the molecular graph of m − BiI 3 has two types of edge sets. e edge partition of the edge set of m − BiI 3 is given in Table 5.

Journal of Chemistry
Proof. From the edge partition of the edge set of m − BiI 3 given in Table 5, we have (1) e first Gourava index for m − BiI 3 is (2) e second Gourava index for m − BiI 3 is  Proof. From the edge partition of the edge set of m − BiI 3 given in Table 5 × n). e molecular graph of the bismuth triiodide sheet BiI 3 (m × n) is shown in Figure 23. It can be observed from Figure 23 that the edge set of the bismuth triiodide sheet BiI 3 (m × n) can be divided into three classes based on the degrees of end vertices as shown in Table 6.

Theorem 11. Let G be the graph of the bismuth triiodide sheet BiI 3 (m × n). en, the first and second Gourava indices are
Proof. From the edge partition of the edge set of the bismuth triiodide sheet BiI 3 (m × n) given in Table 6, we have (2) e second Gourava index for BiI 3 (m × n) is (2) e second hyper-Gourava index for BiI 3 (m × n) is     e algebraic graph of porphyrin dendrimer D n P n is shown in Figure 32. For porphyrin dendrimer D n P n , |V(D n P n )| � 96n − 10 and |E(D n P n )| � 105n − 11. ere are six type of edges in the edge set of porphyrin dendrimer, based on the degree of end vertices. Degree-based partition of edges of porphyrin dendrimer D n P n is given in Table 7.

Gourava Indices for
Theorem 13. Let G be the graph of porphyrin dendrimer D n P n . en, the first and second Gourava indices are Proof. From the edge partition of D n P n given in Table 7, we have (1) e first Gourava index for D n P n is □ Theorem 14. Let G be the graph of porphyrin dendrimer D n P n . en, the first and second hyper-Gourava indices are Proof. From the edge partition of D n P n given in Table 7, we have (1) e first hyper-Gourava index for D n P n is (2) e second hyper-Gourava index for D n P n is
ere are six type of edges in the edge set of porphyrin dendrimer, based on the degree of end vertices. Degreebased partition of edges of propyl ether imine dendrimer (PETIM) is given in Table 8. Proof. From the edge partition of PETIM given in Table 8, we have (1) e first Gourava index for PETIM is (2) e second Gourava index for PETIM is � 18.2 n + 6.n n+1 + 16.2 n+4 − 528. Proof. From the edge partition of PETIM given in Table 8,  we have (1) e first hyper-Gourava index for PETIM is (2) e second hyper-Gourava index for PETIM is   e algebraic graph of zinc-porphyrin dendrimer DPZ n is shown in Figure 34. ere are six type of edges in the edge set of porphyrin dendrimer, based on the degree of end vertices. Degree-based partition of edges of zinc-porphyrin dendrimer DPZ n is given in Table 9.  Table 8: Degree-based edge partition of (PETIM).
Theorem 17. Let G be the graph of zinc-porphyrin dendrimer DPZ n . en, the first and second Gourava indices are Proof. From the edge partition of DPZ n given in Table 9 Proof. From the edge partition of DPZ n given in Table 9, we have (1) e first hyper-Gourava index for DPZ n is (2) e second hyper-Gourava index for DPZ n is 2 Table 10. Proof. From the edge partition of PETAA given in Table 10,  we have (1) e first Gourava index for PETAA is (2) e second Gourava index for PETAA is     Proof. From the edge partition of PETAA given in Table 10    In this section, we will present the graphical comparison of first, second, first hyper-, and second hyper-Gourava indices for porphyrin dendrimer D n P n , propyl ether imine dendrimer (PETIM), zinc-porphyrin dendrimer DPZ n and Poly(E yleneAmidoAmine) dendrimer (PETAA). Figures 36-39 show the all indices are linearly increasing with respect to involved parameters.

Conclusions and Future Works
It is important to calculate topological indices of dendrimers because it is a proved fact that topological indices help to predict many properties without going to the wet lab. ere are more than around 148 topological indices, but none of them can completely describe all properties of a chemical compound. erefore, there is always room to define and study new topological indices. Gourava indices are one step in this direction and are very close to Zagreb indices. Zagreb indices are very well studied by chemists and mathematicians due to their huge applications in chemistry. It is an interesting problem for researchers to study chemical properties and bonds of Gourava indices.

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

Conflicts of Interest
e authors of this paper declare that they have no conflicts of interest.