Abstract

Silicon carbide (SiC), also called carborundum, is a semiconductor containing silicon and carbon. Dendrimers are repetitively branched molecules that are typically symmetric around the core and often adopt a spherical three-dimensional morphology. Bismuth(III) iodide is an inorganic compound with the formula . This gray-black solid is the product of the reaction between bismuth and iodine, which once was of interest in qualitative inorganic analysis. In chemical graph theory, we associate a graph to a compound and compute topological indices that help us in guessing properties of the understudy compound. A topological index is the graph invariant number, calculated from a graph representing a molecule. Most of the proposed topological indices are related either to a vertex adjacency relationship (atom-atom connectivity) in the graph or to topological distances in the graph. In this paper, we aim to compute the first and second Gourava indices and hyper-Gourava indices for silicon carbides, bismuth(III) iodide, and dendrimers.

1. 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). There are three kinds of topological indices, namely, degree-based, distance-based, and surface-based topological indices. Lot of research has been done on degree-based topological indices, for example, see [1–9]. Degree-based topological indices correlate the structure of the molecular compound with its various physical properties, biological activities, and chemical reactivity [10–14]. Boiling point, heat of formation, fracture toughness, strain energy, and rigidity of a molecule are strongly connected to its graphical structure.

The 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 [16–20]. 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 [22–27]. The other oldest topological indices are Zagreb indices defined by Gutman and Trinajstic in [28] and are one of the most studied topological indices [29–33]. Topological indices are helpful in guessing properties of concerned compounds and are used in QSPRs [34–37]. There are more than 148 topological indices in the literature [38–42], but none of them are able to guess all the properties of the concerned compound (together they do it to some extent). Therefore, there is always room to define new topological indices [43]. Recently, in 2017, the first and second Gourava indices [44] were defined as

In the same year, the first and second hyper-Gourava indices [45] have been defined as

Note that , , , and . 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.

2. 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.

3. Gourava Indices

In this section, we present our main computational results. This section consists of three subsections. In Section 3.1, we present results about silicone carbides , , , and . In Section 3.2, we give results about the bismuth triiodide chain and the bismuth triiodide sheet . In Section 3.3, we present results about four dendrimer structures: porphyrin dendrimer , propyl ether imine dendrimer (), zinc-porphyrin dendrimer , and Poly(EThyleneAmidoAmine) dendrimer ().

3.1. 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 SiO₂ 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 , , , and .

3.1.1. Gourava Indices for Silicon Carbide

The molecular graphs of silicon carbide are shown in Figures 1–4, where Figure 1 shows the unit cell of silicone carbide, Figure 2 shows for , , Figure 3 shows for , , and Figure 4 shows for , . The edge partition of the edge set of based on the degree of the end vertex is given in Table 1.

Figure 1 

Unit cell of .

Figure 2 

for , .

Figure 3 

for , .

Figure 4 

for , .

Theorem 1. Let be the graph of silicon carbide . Then, the first and second Gourava indices are(1),(2).

Proof. From the edge partition of given in Table 1, we have(1)The first Gourava index for is(2)The second Gourava index for is

Theorem 2. Let be the graph of silicon carbide . Then, the first and second hyper-Gourava indices are(1),(2).

Proof. From the edge partition of given in Table 1, we have(1)The first hyper-Gourava index for is(2)The second hyper-Gourava index for is

3.1.2. Gourava Indices for Silicon Carbide

The molecular graphs of silicon carbide are shown in Figures 5–8, where Figure 5 shows the unit cell of , Figure 6 shows for , , Figure 7 shows for , , and Figure 8 shows for , . The edge partition of the edge set of based on the degree of the end vertex is given in Table 2.

Figure 5 

Unit cell of .

Figure 6 

for , .

Figure 7 

for , .

Figure 8 

for , .

Theorem 3. Let be the graph of silicon carbide . Then, the first and second Gourava indices are(1),(2).

Proof. From the edge partition of given in Table 2, we have(1)The first Gourava index for is(2)The second Gourava index for is

Theorem 4. Let be the graph of silicon carbide . Then, the first and second hyper-Gourava indices are(1),(2).

Proof. From the edge partition of given in Table 2, we have(1)The first hyper-Gourava index for is(2)The second hyper-Gourava index for is

3.1.3. Gourava Indices for Silicon Carbide

The unit cell of is shown in Figure 9. The 2D lattice graphs of , , and are shown in Figures 10–12, respectively. The edge partition of the edge set of based on the degrees of end vertices is given in Table 3.

Figure 9 

Unit cell.

Figure 10 

.

Figure 11 

.

Figure 12 

.

Theorem 5. Let be the graph of silicon carbide . Then, the first and second Gourava indices are(1),(2).

Proof. From the edge partition of given in Table 3, we have(1)The first Gourava index for is(2)The second Gourava index for is

Theorem 6. Let be the graph of silicon carbide . Then, the first and second hyper-Gourava indices are(1),(2).

Proof. From the edge partition of given in Table 3, we have(1)The first hyper-Gourava index for is(2)The second hyper-Gourava index for is