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

Authors: Qi-Zhao Li; Abaid ur Rehman Virk; Kashif Nazar; Imran Ahmed; Iskander Tlili
Journal: Journal of Chemistry
 (2020)

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## 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 formulaBiI3. 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.


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## Body

## 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(1)GO1G=∑uvϵEGdu+dv+du.dv,GO2G=∑uvϵEGdu+dv.du.dv.In the same year, the first and second hyper-Gourava indices [45] have been defined as(2)HGO1G=∑uvϵEGdu+dv+du.dv2,HGO2G=∑uvϵEGdu+dv.du.dv2.Note thatGO1G=M1G+M2G, GO2G=M1GM2G, HGO1G=H1G+H2G+2M1G+M2G, and HGO2G=H1GH2G. 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 Section3.1, we present results about silicone carbides Si2C3−Ip,q, Si2C3−IIp,q, Si2C3−IIIp,q, and SiC3−IIIp,q. In Section 3.2, we give results about the bismuth triiodide chain m−BiI3 and the bismuth triiodide sheet BiI3m×n. In Section 3.3, we present results about four dendrimer structures: porphyrin dendrimer DnPn, propyl ether imine dendrimer (PETIM), zinc-porphyrin dendrimer DPZn, and Poly(EThyleneAmidoAmine) dendrimer (PETAA).


### 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 Si2C3−Ip,q, Si2C3−IIp,q, Si2C3−IIIp,q, and SiC3−IIIp,q.


#### 3.1.1. Gourava Indices for Silicon CarbideSi2C3−Ip,q

The molecular graphs of silicon carbideSi2C3−Ip,q are shown in Figures 1–4, where Figure 1 shows the unit cell of silicone carbide, Figure 2 shows Si2C3Ip,q for p=4, q=3, Figure 3 shows Si2C3Ip,q for p=4, q=1, and Figure 4 shows Si2C3Ip,q for p=4, q=3. The edge partition of the edge set of Si2C3−Ip,q based on the degree of the end vertex is given in Table 1.Figure 1
Unit cell ofSi2C3Ip,q.Figure 2
Si2C3Ip,q for p=4, q=3.Figure 3
Si2C3Ip,q for p=4, q=1.Figure 4
Si2C3Ip,q for p=4, q=3.Table 1
Degree-based edge partition ofSi2C3Ip,q.
du,dvFrequency1,211,312,2p+2q2,36p−1+8q−13,315pq−9p−13q+7Theorem 1.
LetG be the graph of silicon carbide Si2C3Ip,q. Then, the first and second Gourava indices are(1)
GO1G=225pq−61p−91q+18,(2)
GO2G=810pq−290p−430q+126.Proof.
From the edge partition ofSi2C3Ip,q given in Table 1, we have(1)
The first Gourava index forSi2C3Ip,q is(3)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×21+1+3+1×31+2+2+2×2p+2q+2+3+2×36p+8q−9+3+3+3×315pq−9p−13q+7=225pq−61p−91q+18.(2)
The second Gourava index forSi2C3Ip,q is(4)GO2G=∑uvϵEGdu+dv.du.dv=1+2×1+21+1+3×1+31+2+2×2+2p+2q+2+3×2×36p+8q−9+3+3×3+315pq−9p−13q+7=810pq−290p−430q+126.Theorem 2.
LetG be the graph of silicon carbide Si2C3Ip,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=3375pq−1235p−1829q+566,(2)
HGO2G=43740pq−20588p−30196q+12492.Proof.
From the edge partition ofSi2C3Ip,q given in Table 1, we have(1)
The first hyper-Gourava index forSi2C3Ip,q is(5)HGO1G=∑uv∈EGdu+dv+du.dv2=1×2+1×221+1+3+1×321+2+2+2×22p+2q+2+3+2×326p+8q−9+3+3+3×3215pq−9p−13q+7=3375pq−1235p−1829q+566.(2)
The second hyper-Gourava index forSi2C3Ip,q is(6)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1+221+1+3×1+321+2+2×2+22p+2q+2+3×2+326p+8q−9+3+3×3×3215pq−9p−13q+7=43740pq−20588p−30196q+12492.


#### 3.1.2. Gourava Indices for Silicon CarbideSi2C3−IIp,q

The molecular graphs of silicon carbideSi2C3−IIp,q are shown in Figures 5–8, where Figure 5 shows the unit cell of Si2C3−IIp,q, Figure 6 shows Si2C3−IIp,q for p=3, q=3, Figure 7 shows Si2C3−IIp,q for p=5, q=1, and Figure 8 shows Si2C3−IIp,q for p=5, q=2. The edge partition of the edge set of Si2C3−IIp,q based on the degree of the end vertex is given in Table 2.Figure 5
Unit cell ofSi2C3−IIp,q.Figure 6
Si2C3−IIp,q for p=3, q=3.Figure 7
Si2C3−IIp,q for p=5, q=1.Figure 8
Si2C3−IIp,q for p=5, q=2.Table 2
Degree-based edge partition ofSi2C3−IIp,q.
du,dvFrequency1,221,312,22p+2q2,38p+8q−143,315pq−13p−13q+11Theorem 3.
LetG be the graph of silicon carbide Si2C3−IIp,q. Then, the first and second Gourava indices are(1)
GO1G=225pq−91p−91q+28,(2)
GO2G=810pq−430p−430q+198.Proof.
From the edge partition ofSi2C3−IIp,q given in Table 2, we have(1)
The first Gourava index forSi2C3Ip,q is(7)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×22+1+3+1×31+2+2+2×22p+2q+2+3+2×38p+8q−14+3+3+3×315pq−13p−13q+11=225pq−91p−91q+28.(2)
The second Gourava index forSi2C3−IIp,q is(8)GO2G=∑uvϵEGdu+dv.du.dv=1+2×1×22+1+3×1×31+2+2×2×22p+2q+2+3×2×38p+8q−14+3+3×3×315pq−13p−13q+11=1080pq−680p−680q+396.Theorem 4.
LetG be the graph of silicon carbide Si2C3−IIp,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=3375pq−1829p−1829q+880,(2)
HGO2G=43740pq−30196p−30196q+19692.Proof.
From the edge partition ofSi2C3−IIp,q given in Table 2, we have(1)
The first hyper-Gourava index forSi2C3−IIp,q is(9)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×222+1+3+1×321+2+2+2×222p+2q+2+3+2×328p+8q−14+3+3+3×3215pq−3p−13q+11=3375pq−1829p−1829q+880.(2)
The second hyper-Gourava index forSi2C3−IIp,q is(10)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×222+1+3×1×321+2+2×2×222p+2q+2+3×2×328p+8q−14+3+3×3×3215pq−13p−13q+11=43740pq−30196p−30196q+19692.


#### 3.1.3. Gourava Indices for Silicon CarbideSi2C3−IIIp,q

The unit cell ofSi2C3IIIp,q is shown in Figure 9. The 2D lattice graphs of Si2C3−I5,1, Si2C3−I5,2, and Si2C3−I5,4 are shown in Figures 10–12, respectively. The edge partition of the edge set of Si2C3IIIp,q based on the degrees of end vertices is given in Table 3.Figure 9
Unit cell.Figure 10
Si2C3−I5,1.Figure 11
Si2C3−I5,2.Figure 12
Si2C3−I5,4.Table 3
Degree-based edge partition ofSi2C3IIIp,q.
du,dvFrequency1,322,22q+22,38p+8q−123,315pq−10p−13q+8Theorem 5.
LetG be the graph of silicon carbide Si2C3IIIp,q. Then, the first and second Gourava indices are(1)
GO1G=225pq−62p−91q+18,(2)
GO2G=810pq−300p−430q+128.Proof.
From the edge partition ofSi2C3IIIp,q given in Table 3, we have(1)
The first Gourava index forSi2C3IIIp,q is(11)GO1G=∑uvϵEGdu+dv+du.dv=1×31+2+2+2×22p+2q+2+3+2×38p+8q−14+3+3+3×315pq−13p−13q+11=225pq−62p−91q+18.(2)
The second Gourava index forSi2C3IIIp,q is(12)GO2G=∑uvϵEGdu+dv.du.dv=1+3×1×31+2+2×2×22p+2q+2+3×2×38p+8q−14+3+3×3×315pq−13p−13q+11=810pq−300p−430q+128.Theorem 6.
LetG be the graph of silicon carbide Si2C3IIIp,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=3375pq−1282p−1829q+500,(2)
HGO2G=43740pq−21960p−30196q+13328.Proof.
From the edge partition ofSi2C3IIIp,q given in Table 3, we have(1)
The first hyper-Gourava index forSi2C3IIIp,q is(13)HGO1G=∑uvϵEGdu+dv+du.dv2=1+3+1×321+2+2+2×222p+2q+2+3+2×328p+8q−14+3+3+3×3215pq−3p−13q+11=3375pq−1282p−1829q+500.(2)
The second hyper-Gourava index forSi2C3IIIp,q is(14)HGO2G=∑uvϵEGdu+dv.du.dv2=1+3×1×321+2+2×2×222p+2q+2+3×2×328p+8q−14+3+3×3×3215pq−13p−13q+11=43740pq−21960p−30196q+13328.


#### 3.1.4. Gourava Indices for Silicon CarbideSiC3−IIIp,q

The unit cell ofSiC3−IIIp,q is shown in Figure 13. The 2D lattice graphs of SiC3−III5,1, SiC3−III5,2, and SiC3−III5,4 are shown in Figures 14–16, respectively. The edge partition of the edge set of SiC3−IIIp,q based on the degrees of end vertices is given in Table 4.Figure 13
Unit cell.Figure 14
SiC3−III5,1.Figure 15
SiC3−III5,2.Figure 16
SiC3−III5,4.Table 4
Degree-based edge partition ofSiC3IIIp,q.
du,dvFrequency1,221,312,23p+2q−32,36p+4q−83,312pq−12p−8q+8Theorem 7.
LetG be the graph of silicon carbide SiC3IIIp,q. Then, the first and second Gourava indices are(1)
GO1G=480pq−90p−60q+25,(2)
GO2G=648pq−420p−280q+168.Proof.
From the edge partition of the edge set ofSiC3IIIp,q given in Table 4, we have(1)
The first Gourava index forSiC3IIIp,q is(15)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×22+1+3+1×31+2+2+2×23p+2q−3+2+3+2×36p+4q−8+3+3+3×312pq−12p−8q+8=480pq−90p−60q+25.(2)
The second Gourava index forSiC3IIIp,q is(16)GO2G=∑uvϵEGdu+dv.du.dv=1+2×1×22+1+3×1×31+2+2×2×23p+2q−3+2+3×2×36p+4q−8+3+3×3×312pq−12p−8q+8=648pq−420p−280q+168.Theorem 8.
LetG be the graph of silicon carbide SiC3IIIp,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=2700pq−1782p−118q+739,(2)
HGO2G=34992pq−28824p−19216q+15576.Proof.
From the edge partition of the edge set ofSiC3IIIp,q given in Table 4, we have(1)
The first hyper-Gourava index forSiC3IIIp,q is(17)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×222+1+3+1×321+2+2+2×222p+2q+2+3+2×328p+8q−14+3+3+3×3215pq−3p−13q+11=2700pq−1782p−118q+739.(2)
The second hyper-Gourava index forSiC3IIIp,q is(18)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×222+1+3×1×321+2+2×2×222p+2q+2+3×2×328p+8q−14+3+3×3×3215pq−13p−13q+11=34992pq−28824p−19216q+15576.


#### 3.1.5. Graphical Comparison of Results of Silicone Carbides

In Figures17–20, we can observe that the behavior of all indices is exponentially increasing with respect to the involved parameters.Figure 17
Comparison of the first Gourava index.Figure 18
Comparison of the second Gourava index.Figure 19
Comparison of the first hyper-Gourava index.Figure 20
Comparison of the second hyper-Gourava index.Codes for plotting the first and second Gourava indices for silicon carbideSi2C3Ip,r are given as follows:(19)plot3d225∗p∗q−61∗p−91∗q+18,p=0..1,q=0..1,colour=red,plot3d810∗p∗q−290∗p−430∗q+126,p=0..1,q=0..1,colour=green.


### 3.2. Gourava Indices for Bismuth Triiodide

BiI3 is an inorganic compound which is the result of the reaction between iodine and bismuth, which inspired the enthusiasm for subjective inorganic investigations [46]. BiI3 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 BiI3 gemstones are considered to be a three-layered stack structure in which a plane of bismuth atoms is sandwiched between iodide particle planes to form a continuous plane [49]. The periodic superposition of the three layers forms diamond-shaped BiI3 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 BiI3 has been integrated in [46]. We referred to [6] for the topological study of bismuth triiodide.


#### 3.2.1. Bismuth Triiodide Chainm−BiI3

The molecular graph of the unit cell ofm−BiI3 is shown in Figure 21. From Figure 22, we can see that the molecular graph of m−BiI3 has two types of edge sets. The edge partition of the edge set of m−BiI3 is given in Table 5.Figure 21
Unit cell (bismuth triiodide).Figure 22
The chain form=3 (bismuth triiodide).Table 5
Degree-based edge partition ofm−Bil3m,n of end vertices of each edge.
du,dvFrequency1,64m+82,620m+4Theorem 9.
LetG be the graph of the bismuth triiodide chain m−BiI3. Then, the first and second Gourava indices are(1)
GO1G=452m+184,(2)
GO2G=2088m+720.Proof.
From the edge partition of the edge set ofm−BiI3 given in Table 5, we have(1)
The first Gourava index form−BiI3 is(20)GO1G=∑uvϵEGdu+dv+du.dv=1+6+1×64m+8+2+6+2×620m+4=452m+184.(2)
The second Gourava index form−BiI3 is(21)GO2G=∑uvϵEGdu+dv.du.dv=1+6×1×64m+8+2+6×2×620m+4=2088m+720.Theorem 10.
LetG be the graph of the bismuth triiodide chain m−BiI3. Then, the first and second hyper-Gourava indices are(1)
HGO1G=8676m+2952,(2)
HGO2G=191376m+93376.Proof.
From the edge partition of the edge set ofm−BiI3 given in Table 5, we have(1)
The first hyper-Gourava index form−BiI3 is(22)HGO1G=∑uvϵEGdu+dv+du.dv2=1+6+1×624m+8+2+6+2×6220m+4=8676m+2952.(2)
The second hyper-Gourava index form−BiI3 is(23)HGO2G=∑uvϵEGdu+dv.du.dv2=1+6×1×624m+8+2+6×2×6220m+4=191376m+93376.


#### 3.2.2. Bismuth Triiodide SheetBiI3m×n

The molecular graph of the bismuth triiodide sheetBiI3m×n is shown in Figure 23. It can be observed from Figure 23 that the edge set of the bismuth triiodide sheet BiI3m×n can be divided into three classes based on the degrees of end vertices as shown in Table 6.Figure 23
Bismuth triiodide sheet.Table 6
Degree-based edge partition ofBiI3m×n.
du,dvFrequency1,64m+n+12,643mn+2m+2n−13,66nm−1Theorem 11.
LetG be the graph of the bismuth triiodide sheet BiI3m×n. Then, the first and second Gourava indices are(1)
GO1G=402mn+212m+50n−28,(2)
GO2G=2124mn+936m−36n−216.Proof.
From the edge partition of the edge set of the bismuth triiodide sheetBiI3m×n given in Table 6, we have(1)
The first Gourava index forBiI3m×n is(24)GO1G=∑uvϵEGdu+dv+du.dv=1+6+1×64m+4n+4+2+6+2×612mn+8m+8n−4+3+6+3×66mn−6n=402mn+212m+50n−28.(2)
The second Gourava index forBiI3m×n is(25)GO2G=∑uvϵEGdu+dv.du.dv=1+6×1×64m+4n+4+2+6×2×612mn+8m+8n−4+3+6×3×66mn−6n=2124mn+936m−36n−216.Theorem 12.
LetG be the graph of the bismuth triiodide sheet BiI3m×n. Then, the first and second hyperGourava indices are(1)
HGO1G=9174mn+3876m−498n−924,(2)
HGO2G=268056mn+80784m−143036n−29808.Proof.
From the edge partition of the edge set of the bismuth triiodide sheetBiI3m×n given in Table 6, we have(1)
The first hyper-Gourava index forBiI3m×n is(26)HGO1G=∑uvϵEGdu+dv+du.dv2=1+6+1×624m+4n+4+2+6+2×6212mn+8m+8n−4+3+6+3×626mn−6n=9174mn+3876m−498n−924.(2)
The second hyper-Gourava index forBiI3m×n is(27)HGO2G=∑uvϵEGdu+dv.du.dv2=1+6×1×624m+4n+4+2+6×2×6212mn+8m+8n−4+3+6×3×626mn−6n=268056mn+80784m−143036n−29808.


### 3.3. Graphical Representation

Graphical representation of computed topological indices for the bismuth triiodide chain is shown in Figures24–27, and the graphical representation of the bismuth triiodide sheet is shown in Figures 28–31.Figure 24
Plots of the first Gourava index form−Bil3.Figure 25
Plots of the second Gourava index form−Bil3.Figure 26
Plots of the first hyper-Gourava index form−Bil3.Figure 27
Plots of the second hyper-Gourava index form−Bil3.Figure 28
Plots of the first Gourava index forBil3m×n.Figure 29
Plots of the second Gourava index forBil3m×n.Figure 30
Plots of the first hyper-Gourava index forBil3m×n.Figure 31
Plots of the second hyper-Gourava index forBil3m×n.


### 3.4. Gourava Indices for Dendrimers

In the medication mathematical model, the structure of the drug is addressed as an undirected graph, where each vertex exhibits a molecule and each edge addresses a bond between atoms. A huge number of new drugs have been made each year. From this time forward, it asks for a giant measure of work to choose the pharmacological compound and organic qualities of these new drugs, and such remaining tasks at hand end up being progressively specific and grouped. It requires enough reagent rigging and accomplices to test the exhibitions and the responses of new drugs. Nevertheless, in cut down poor countries and locales (for instance, certain urban networks and countries in South America, Southeast Asia, Africa, and India), there is no sufficient money to settle reagents and apparatus which can be used to gauge the biochemical properties. For topological study of dendrimers, we refer [53–66].


#### 3.4.1. Gourava Indices of Porphyrin DendrimerDnPn

The algebraic graph of porphyrin dendrimerDnPn is shown in Figure 32. For porphyrin dendrimer DnPn, VDnPn=96n−10 and EDnPn=105n−11. There 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 DnPn is given in Table 7.Figure 32
Porphyrin dendrimerDnPn.Table 7
Degree-based edge partition ofDnPn.
du,dvFrequency1,32n1,4124n2,210n−52,348n−63,313n3,48nTheorem 13.
LetG be the graph of porphyrin dendrimer DnPn. Then, the first and second Gourava indices are(1)
GO1G=1169n−106,(2)
GO2G=3478n−260.Proof.
From the edge partition ofDnPn given in Table 7, we have(1)
The first Gourava index forDnPn is(28)GO1G=∑uvϵEGdu+dv+du.dv=1+3+1×32n+1+4+1×424n+2+2+2×210n−5+2+3+2×348n−6+3+3+3×313n+3+4+3×48n=1169n−106.(2)
The second Gourava index forDnPn is(29)GO2DnPn=∑uvϵEGdu+dv.du.dv=1+3×1×32n+1+4×1×424n+2+2×2×210n−5+2+3×2×348n−6+3+3×3×313n+3+4×3×48n=3478n−260.Theorem 14.
LetG be the graph of porphyrin dendrimer DnPn. Then, the first and second hyper-Gourava indices are(1)
HGO1G=13727n−1046,(2)
HGO2G=150004n−6680.Proof.
From the edge partition ofDnPn given in Table 7, we have(1)
The first hyper-Gourava index forDnPn is(30)HGO1G=∑uvϵEGdu+dv+du.dv2=1+3+1×322n+1+4+1×4224n+2+2+2×2210n−5+2+3+2×3248n−6+3+3+3×3213n+3+4+3×428n=13727n−1046.(2)
The second hyper-Gourava index forDnPn is(31)HGO2G=∑uvϵEGdu+dv.du.dv2=1+3×1×322n+1+4×1×4224n+2+2×2×2210n−5+2+3×2×3248n−6+3+3×3×3213n+3+4×3×48n=150004n−6680.


#### 3.4.2. Propyl Ether Imine Dendrimer (PETIM)

The algebraic graph of propyl ether imine dendrimer (PETIM) is shown in Figure 33. For propyl ether imine dendrimer (PETIM), VPETIM=24.2n−23 and EPETIM=24.2n−24. There 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 propyl ether imine dendrimer (PETIM) is given in Table 8.Figure 33
Propyl ether imine dendrimer (PETIM).Table 8
Degree-based edge partition of (PETIM).
du,dvFrequency1,22n+12,22n+4−182,36.2n−6Theorem 15.
LetG be the graph of propyl ether imine dendrimer (PETIM). Then, the first and second Gourava indices are(1)
GO1G=66.2n+5nn+1+8.2n+4−232,(2)
GO2G=18.2n+6.nn+1+16.2n+4−528.Proof.
From the edge partition ofPETIM given in Table 8, we have(1)
The first Gourava index forPETIM is(32)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×22n+1+2+2+2×22n+4−18+2+3+2×36.2n−6=66.2n+5nn+1+8.2n+4−232.(2)
The second Gourava index forPETIM is(33)GO2PETIM=∑uvϵEGdu+dv.du.dv=1+2×1×22n+1+2+2×2×22n+4−18+2+3×2×36.2n−6=18.2n+6.nn+1+16.2n+4−528.Theorem 16.
LetG be the graph of propyl ether imine dendrimer (PETIM). Then, the first and second hyper-Gourava indices are(1)
HGO1G=726.2n+25.nn+1+64.2n+4−2120,(2)
HGO2G=5400.2n+36.nn+1+256.2n+4−11808.Proof.
From the edge partition ofPETIM given in Table 8, we have(1)
The first hyper-Gourava index forPETIM is(34)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×222n+1+2+2+2×222n+4−18+2+3+2×326.2n−6=726.2n+25.nn+1+64.2n+4−2120.(2)
The second hyper-Gourava index forPETIM is(35)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×222n+1+2+2×2×222n+4−18+2+3×2×36.2n−6=5400.2n+36.nn+1+256.2n+4−11808.


#### 3.4.3. Zinc-Porphyrin DendrimerDPZn

The algebraic graph of zinc-porphyrin dendrimerDPZn is shown in Figure 34. There 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 DPZn is given in Table 9.Figure 34
Zinc-porphyrin dendrimerDPZn.Table 9
Degree-based edge partition ofDPZn.
du,dvFrequency2,216.2n−42,340.2n−163,38.2n−163,44Theorem 17.
LetG be the graph of zinc-porphyrin dendrimer DPZn. Then, the first and second Gourava indices are(1)
GO1G=688.2n−372,(2)
GO2G=1888.2n−1072.Proof.
From the edge partition ofDPZn given in Table 9, we have(1)
The first Gourava index forDPZn is(36)GO1G=∑uvϵEGdu+dv+du.dv=2+2+2×216.2n−4+2+3+2×340.2n−16+3+3+3×38.2n−16+3+4+3×44=688.2n−372.(2)
The second Gourava index forDPZn is(37)GO2G=∑uvϵEGdu+dv.du.dv=2+2×2×216.2n−4+2+3×2×340.2n−16+3+3×3×38.2n−16+3+4×3×44=1888.2n−1072.Theorem 18.
LetG be the graph of zinc-porphyrin dendrimer DPZn. Then, the first and second hyper-Gourava indices are(1)
HGO1G=7664nn−5716,(2)
HGO2G=63424.2n−33856.Proof.
From the edge partition ofDPZn given in Table 9, we have(1)
The first hyper-Gourava index forDPZn is(38)HGO1G=∑uvϵEGdu+dv+du.dv2=2+2+2×2216.2n−4+2+3+2×3240.2n−16+3+3+3×328.2n−16+3+4+3×424=7664nn−5716.(2)
The second hyper-Gourava index forDPZn is(39)HGO2G=∑uvϵEGdu+dv.du.dv2=2+2×2×2216.2n−4+2+3×2×3240.2n−16+3+3×3×328.2n−16+3+4×3×424=63424.2n−33856.


#### 3.4.4. Poly(EThyleneAmidoAmine) Dendrimer (PETAA)

The algebraic graph of Poly(EThyleneAmidoAmine) dendrimer (PETAA) is shown in Figure 35. For Poly(EThyleneAmidoAmine) dendrimer (PETAA), VPETAA=44.2n−18 and EPETAA=44.2n−19. There 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 Poly(EThyleneAmidoAmine) dendrimer (PETAA) is given in Table 10.Figure 35
Poly(EThylene amide amine) dendrimer (PETAA).Table 10
Degree-based edge partition of (PETAA).
du,dvFrequency1,24.2n1,34.2n−22,216.2n−82,320.2n−9Theorem 19.
LetG be the graph of Poly(EThylene Amide Amine) dendrimer (PETAA). Then, the first and second Gourava indices are(1)
GO1G=369.2n−177,(2)
GO2G=928.2n−422.Proof.
From the edge partition ofPETAA given in Table 10, we have(1)
The first Gourava index forPETAA is(40)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×24.2n+1+3+1×34.2n−2+2+2+2×216.2n−8+2+3+2×320.2n−9=369.2n−177.(2)
The second Gourava index forPETAA is(41)GO2PETAA=∑uvϵEGdu+dv.du.dv=1+2×1×24.2n+1+3×1×34.2n−2+2+2×2×216.2n−8+2+3×2×320.2n−9=928.2n−422.Theorem 20.
LetG be the graph of Poly(EThylene Amide Amine) dendrimer (PETAA). Then, the first and second hyper-Gourava indices are(1)
HGO1G=3740.2n−1699,(2)
HGO2G=22816.2n−10436.Proof.
From the edge partition ofPETAA given in Table 10, we have(1)
The first hyper-Gourava index forPETAA is(42)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×224.2n+1+3+1×324.2n−2+2+2+2×2216.2n−8+2+3+2×3220.2n−9=3740.2n−1699.(2)
The second hyper-Gourava index forPETAA is(43)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×224.2n+1+3×1×324.2n−2+2+2×2×2216.2n−8+2+3×2×3220.2n−9=22816.2n−10436.


#### 3.4.5. Graphical Comparison

In this section, we will present the graphical comparison of first, second, first hyper-, and second hyper-Gourava indices for porphyrin dendrimerDnPn, propyl ether imine dendrimer (PETIM), zinc-porphyrin dendrimer DPZn and Poly(EThyleneAmidoAmine) dendrimer (PETAA). Figures 36–39 show the all indices are linearly increasing with respect to involved parameters.Figure 36
First Gourava index.Figure 37
Second Gourava index.Figure 38
First hyper-Gourava index.Figure 39
Second hyper-Gourava index.


## 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 Si2C3−Ip,q, Si2C3−IIp,q, Si2C3−IIIp,q, and SiC3−IIIp,q.


### 3.1.1. Gourava Indices for Silicon CarbideSi2C3−Ip,q

The molecular graphs of silicon carbideSi2C3−Ip,q are shown in Figures 1–4, where Figure 1 shows the unit cell of silicone carbide, Figure 2 shows Si2C3Ip,q for p=4, q=3, Figure 3 shows Si2C3Ip,q for p=4, q=1, and Figure 4 shows Si2C3Ip,q for p=4, q=3. The edge partition of the edge set of Si2C3−Ip,q based on the degree of the end vertex is given in Table 1.Figure 1
Unit cell ofSi2C3Ip,q.Figure 2
Si2C3Ip,q for p=4, q=3.Figure 3
Si2C3Ip,q for p=4, q=1.Figure 4
Si2C3Ip,q for p=4, q=3.Table 1
Degree-based edge partition ofSi2C3Ip,q.
du,dvFrequency1,211,312,2p+2q2,36p−1+8q−13,315pq−9p−13q+7Theorem 1.
LetG be the graph of silicon carbide Si2C3Ip,q. Then, the first and second Gourava indices are(1)
GO1G=225pq−61p−91q+18,(2)
GO2G=810pq−290p−430q+126.Proof.
From the edge partition ofSi2C3Ip,q given in Table 1, we have(1)
The first Gourava index forSi2C3Ip,q is(3)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×21+1+3+1×31+2+2+2×2p+2q+2+3+2×36p+8q−9+3+3+3×315pq−9p−13q+7=225pq−61p−91q+18.(2)
The second Gourava index forSi2C3Ip,q is(4)GO2G=∑uvϵEGdu+dv.du.dv=1+2×1+21+1+3×1+31+2+2×2+2p+2q+2+3×2×36p+8q−9+3+3×3+315pq−9p−13q+7=810pq−290p−430q+126.Theorem 2.
LetG be the graph of silicon carbide Si2C3Ip,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=3375pq−1235p−1829q+566,(2)
HGO2G=43740pq−20588p−30196q+12492.Proof.
From the edge partition ofSi2C3Ip,q given in Table 1, we have(1)
The first hyper-Gourava index forSi2C3Ip,q is(5)HGO1G=∑uv∈EGdu+dv+du.dv2=1×2+1×221+1+3+1×321+2+2+2×22p+2q+2+3+2×326p+8q−9+3+3+3×3215pq−9p−13q+7=3375pq−1235p−1829q+566.(2)
The second hyper-Gourava index forSi2C3Ip,q is(6)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1+221+1+3×1+321+2+2×2+22p+2q+2+3×2+326p+8q−9+3+3×3×3215pq−9p−13q+7=43740pq−20588p−30196q+12492.


### 3.1.2. Gourava Indices for Silicon CarbideSi2C3−IIp,q

The molecular graphs of silicon carbideSi2C3−IIp,q are shown in Figures 5–8, where Figure 5 shows the unit cell of Si2C3−IIp,q, Figure 6 shows Si2C3−IIp,q for p=3, q=3, Figure 7 shows Si2C3−IIp,q for p=5, q=1, and Figure 8 shows Si2C3−IIp,q for p=5, q=2. The edge partition of the edge set of Si2C3−IIp,q based on the degree of the end vertex is given in Table 2.Figure 5
Unit cell ofSi2C3−IIp,q.Figure 6
Si2C3−IIp,q for p=3, q=3.Figure 7
Si2C3−IIp,q for p=5, q=1.Figure 8
Si2C3−IIp,q for p=5, q=2.Table 2
Degree-based edge partition ofSi2C3−IIp,q.
du,dvFrequency1,221,312,22p+2q2,38p+8q−143,315pq−13p−13q+11Theorem 3.
LetG be the graph of silicon carbide Si2C3−IIp,q. Then, the first and second Gourava indices are(1)
GO1G=225pq−91p−91q+28,(2)
GO2G=810pq−430p−430q+198.Proof.
From the edge partition ofSi2C3−IIp,q given in Table 2, we have(1)
The first Gourava index forSi2C3Ip,q is(7)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×22+1+3+1×31+2+2+2×22p+2q+2+3+2×38p+8q−14+3+3+3×315pq−13p−13q+11=225pq−91p−91q+28.(2)
The second Gourava index forSi2C3−IIp,q is(8)GO2G=∑uvϵEGdu+dv.du.dv=1+2×1×22+1+3×1×31+2+2×2×22p+2q+2+3×2×38p+8q−14+3+3×3×315pq−13p−13q+11=1080pq−680p−680q+396.Theorem 4.
LetG be the graph of silicon carbide Si2C3−IIp,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=3375pq−1829p−1829q+880,(2)
HGO2G=43740pq−30196p−30196q+19692.Proof.
From the edge partition ofSi2C3−IIp,q given in Table 2, we have(1)
The first hyper-Gourava index forSi2C3−IIp,q is(9)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×222+1+3+1×321+2+2+2×222p+2q+2+3+2×328p+8q−14+3+3+3×3215pq−3p−13q+11=3375pq−1829p−1829q+880.(2)
The second hyper-Gourava index forSi2C3−IIp,q is(10)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×222+1+3×1×321+2+2×2×222p+2q+2+3×2×328p+8q−14+3+3×3×3215pq−13p−13q+11=43740pq−30196p−30196q+19692.


### 3.1.3. Gourava Indices for Silicon CarbideSi2C3−IIIp,q

The unit cell ofSi2C3IIIp,q is shown in Figure 9. The 2D lattice graphs of Si2C3−I5,1, Si2C3−I5,2, and Si2C3−I5,4 are shown in Figures 10–12, respectively. The edge partition of the edge set of Si2C3IIIp,q based on the degrees of end vertices is given in Table 3.Figure 9
Unit cell.Figure 10
Si2C3−I5,1.Figure 11
Si2C3−I5,2.Figure 12
Si2C3−I5,4.Table 3
Degree-based edge partition ofSi2C3IIIp,q.
du,dvFrequency1,322,22q+22,38p+8q−123,315pq−10p−13q+8Theorem 5.
LetG be the graph of silicon carbide Si2C3IIIp,q. Then, the first and second Gourava indices are(1)
GO1G=225pq−62p−91q+18,(2)
GO2G=810pq−300p−430q+128.Proof.
From the edge partition ofSi2C3IIIp,q given in Table 3, we have(1)
The first Gourava index forSi2C3IIIp,q is(11)GO1G=∑uvϵEGdu+dv+du.dv=1×31+2+2+2×22p+2q+2+3+2×38p+8q−14+3+3+3×315pq−13p−13q+11=225pq−62p−91q+18.(2)
The second Gourava index forSi2C3IIIp,q is(12)GO2G=∑uvϵEGdu+dv.du.dv=1+3×1×31+2+2×2×22p+2q+2+3×2×38p+8q−14+3+3×3×315pq−13p−13q+11=810pq−300p−430q+128.Theorem 6.
LetG be the graph of silicon carbide Si2C3IIIp,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=3375pq−1282p−1829q+500,(2)
HGO2G=43740pq−21960p−30196q+13328.Proof.
From the edge partition ofSi2C3IIIp,q given in Table 3, we have(1)
The first hyper-Gourava index forSi2C3IIIp,q is(13)HGO1G=∑uvϵEGdu+dv+du.dv2=1+3+1×321+2+2+2×222p+2q+2+3+2×328p+8q−14+3+3+3×3215pq−3p−13q+11=3375pq−1282p−1829q+500.(2)
The second hyper-Gourava index forSi2C3IIIp,q is(14)HGO2G=∑uvϵEGdu+dv.du.dv2=1+3×1×321+2+2×2×222p+2q+2+3×2×328p+8q−14+3+3×3×3215pq−13p−13q+11=43740pq−21960p−30196q+13328.


### 3.1.4. Gourava Indices for Silicon CarbideSiC3−IIIp,q

The unit cell ofSiC3−IIIp,q is shown in Figure 13. The 2D lattice graphs of SiC3−III5,1, SiC3−III5,2, and SiC3−III5,4 are shown in Figures 14–16, respectively. The edge partition of the edge set of SiC3−IIIp,q based on the degrees of end vertices is given in Table 4.Figure 13
Unit cell.Figure 14
SiC3−III5,1.Figure 15
SiC3−III5,2.Figure 16
SiC3−III5,4.Table 4
Degree-based edge partition ofSiC3IIIp,q.
du,dvFrequency1,221,312,23p+2q−32,36p+4q−83,312pq−12p−8q+8Theorem 7.
LetG be the graph of silicon carbide SiC3IIIp,q. Then, the first and second Gourava indices are(1)
GO1G=480pq−90p−60q+25,(2)
GO2G=648pq−420p−280q+168.Proof.
From the edge partition of the edge set ofSiC3IIIp,q given in Table 4, we have(1)
The first Gourava index forSiC3IIIp,q is(15)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×22+1+3+1×31+2+2+2×23p+2q−3+2+3+2×36p+4q−8+3+3+3×312pq−12p−8q+8=480pq−90p−60q+25.(2)
The second Gourava index forSiC3IIIp,q is(16)GO2G=∑uvϵEGdu+dv.du.dv=1+2×1×22+1+3×1×31+2+2×2×23p+2q−3+2+3×2×36p+4q−8+3+3×3×312pq−12p−8q+8=648pq−420p−280q+168.Theorem 8.
LetG be the graph of silicon carbide SiC3IIIp,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=2700pq−1782p−118q+739,(2)
HGO2G=34992pq−28824p−19216q+15576.Proof.
From the edge partition of the edge set ofSiC3IIIp,q given in Table 4, we have(1)
The first hyper-Gourava index forSiC3IIIp,q is(17)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×222+1+3+1×321+2+2+2×222p+2q+2+3+2×328p+8q−14+3+3+3×3215pq−3p−13q+11=2700pq−1782p−118q+739.(2)
The second hyper-Gourava index forSiC3IIIp,q is(18)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×222+1+3×1×321+2+2×2×222p+2q+2+3×2×328p+8q−14+3+3×3×3215pq−13p−13q+11=34992pq−28824p−19216q+15576.


### 3.1.5. Graphical Comparison of Results of Silicone Carbides

In Figures17–20, we can observe that the behavior of all indices is exponentially increasing with respect to the involved parameters.Figure 17
Comparison of the first Gourava index.Figure 18
Comparison of the second Gourava index.Figure 19
Comparison of the first hyper-Gourava index.Figure 20
Comparison of the second hyper-Gourava index.Codes for plotting the first and second Gourava indices for silicon carbideSi2C3Ip,r are given as follows:(19)plot3d225∗p∗q−61∗p−91∗q+18,p=0..1,q=0..1,colour=red,plot3d810∗p∗q−290∗p−430∗q+126,p=0..1,q=0..1,colour=green.


## 3.1.1. Gourava Indices for Silicon CarbideSi2C3−Ip,q

The molecular graphs of silicon carbideSi2C3−Ip,q are shown in Figures 1–4, where Figure 1 shows the unit cell of silicone carbide, Figure 2 shows Si2C3Ip,q for p=4, q=3, Figure 3 shows Si2C3Ip,q for p=4, q=1, and Figure 4 shows Si2C3Ip,q for p=4, q=3. The edge partition of the edge set of Si2C3−Ip,q based on the degree of the end vertex is given in Table 1.Figure 1
Unit cell ofSi2C3Ip,q.Figure 2
Si2C3Ip,q for p=4, q=3.Figure 3
Si2C3Ip,q for p=4, q=1.Figure 4
Si2C3Ip,q for p=4, q=3.Table 1
Degree-based edge partition ofSi2C3Ip,q.
du,dvFrequency1,211,312,2p+2q2,36p−1+8q−13,315pq−9p−13q+7Theorem 1.
LetG be the graph of silicon carbide Si2C3Ip,q. Then, the first and second Gourava indices are(1)
GO1G=225pq−61p−91q+18,(2)
GO2G=810pq−290p−430q+126.Proof.
From the edge partition ofSi2C3Ip,q given in Table 1, we have(1)
The first Gourava index forSi2C3Ip,q is(3)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×21+1+3+1×31+2+2+2×2p+2q+2+3+2×36p+8q−9+3+3+3×315pq−9p−13q+7=225pq−61p−91q+18.(2)
The second Gourava index forSi2C3Ip,q is(4)GO2G=∑uvϵEGdu+dv.du.dv=1+2×1+21+1+3×1+31+2+2×2+2p+2q+2+3×2×36p+8q−9+3+3×3+315pq−9p−13q+7=810pq−290p−430q+126.Theorem 2.
LetG be the graph of silicon carbide Si2C3Ip,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=3375pq−1235p−1829q+566,(2)
HGO2G=43740pq−20588p−30196q+12492.Proof.
From the edge partition ofSi2C3Ip,q given in Table 1, we have(1)
The first hyper-Gourava index forSi2C3Ip,q is(5)HGO1G=∑uv∈EGdu+dv+du.dv2=1×2+1×221+1+3+1×321+2+2+2×22p+2q+2+3+2×326p+8q−9+3+3+3×3215pq−9p−13q+7=3375pq−1235p−1829q+566.(2)
The second hyper-Gourava index forSi2C3Ip,q is(6)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1+221+1+3×1+321+2+2×2+22p+2q+2+3×2+326p+8q−9+3+3×3×3215pq−9p−13q+7=43740pq−20588p−30196q+12492.


## 3.1.2. Gourava Indices for Silicon CarbideSi2C3−IIp,q

The molecular graphs of silicon carbideSi2C3−IIp,q are shown in Figures 5–8, where Figure 5 shows the unit cell of Si2C3−IIp,q, Figure 6 shows Si2C3−IIp,q for p=3, q=3, Figure 7 shows Si2C3−IIp,q for p=5, q=1, and Figure 8 shows Si2C3−IIp,q for p=5, q=2. The edge partition of the edge set of Si2C3−IIp,q based on the degree of the end vertex is given in Table 2.Figure 5
Unit cell ofSi2C3−IIp,q.Figure 6
Si2C3−IIp,q for p=3, q=3.Figure 7
Si2C3−IIp,q for p=5, q=1.Figure 8
Si2C3−IIp,q for p=5, q=2.Table 2
Degree-based edge partition ofSi2C3−IIp,q.
du,dvFrequency1,221,312,22p+2q2,38p+8q−143,315pq−13p−13q+11Theorem 3.
LetG be the graph of silicon carbide Si2C3−IIp,q. Then, the first and second Gourava indices are(1)
GO1G=225pq−91p−91q+28,(2)
GO2G=810pq−430p−430q+198.Proof.
From the edge partition ofSi2C3−IIp,q given in Table 2, we have(1)
The first Gourava index forSi2C3Ip,q is(7)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×22+1+3+1×31+2+2+2×22p+2q+2+3+2×38p+8q−14+3+3+3×315pq−13p−13q+11=225pq−91p−91q+28.(2)
The second Gourava index forSi2C3−IIp,q is(8)GO2G=∑uvϵEGdu+dv.du.dv=1+2×1×22+1+3×1×31+2+2×2×22p+2q+2+3×2×38p+8q−14+3+3×3×315pq−13p−13q+11=1080pq−680p−680q+396.Theorem 4.
LetG be the graph of silicon carbide Si2C3−IIp,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=3375pq−1829p−1829q+880,(2)
HGO2G=43740pq−30196p−30196q+19692.Proof.
From the edge partition ofSi2C3−IIp,q given in Table 2, we have(1)
The first hyper-Gourava index forSi2C3−IIp,q is(9)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×222+1+3+1×321+2+2+2×222p+2q+2+3+2×328p+8q−14+3+3+3×3215pq−3p−13q+11=3375pq−1829p−1829q+880.(2)
The second hyper-Gourava index forSi2C3−IIp,q is(10)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×222+1+3×1×321+2+2×2×222p+2q+2+3×2×328p+8q−14+3+3×3×3215pq−13p−13q+11=43740pq−30196p−30196q+19692.


## 3.1.3. Gourava Indices for Silicon CarbideSi2C3−IIIp,q

The unit cell ofSi2C3IIIp,q is shown in Figure 9. The 2D lattice graphs of Si2C3−I5,1, Si2C3−I5,2, and Si2C3−I5,4 are shown in Figures 10–12, respectively. The edge partition of the edge set of Si2C3IIIp,q based on the degrees of end vertices is given in Table 3.Figure 9
Unit cell.Figure 10
Si2C3−I5,1.Figure 11
Si2C3−I5,2.Figure 12
Si2C3−I5,4.Table 3
Degree-based edge partition ofSi2C3IIIp,q.
du,dvFrequency1,322,22q+22,38p+8q−123,315pq−10p−13q+8Theorem 5.
LetG be the graph of silicon carbide Si2C3IIIp,q. Then, the first and second Gourava indices are(1)
GO1G=225pq−62p−91q+18,(2)
GO2G=810pq−300p−430q+128.Proof.
From the edge partition ofSi2C3IIIp,q given in Table 3, we have(1)
The first Gourava index forSi2C3IIIp,q is(11)GO1G=∑uvϵEGdu+dv+du.dv=1×31+2+2+2×22p+2q+2+3+2×38p+8q−14+3+3+3×315pq−13p−13q+11=225pq−62p−91q+18.(2)
The second Gourava index forSi2C3IIIp,q is(12)GO2G=∑uvϵEGdu+dv.du.dv=1+3×1×31+2+2×2×22p+2q+2+3×2×38p+8q−14+3+3×3×315pq−13p−13q+11=810pq−300p−430q+128.Theorem 6.
LetG be the graph of silicon carbide Si2C3IIIp,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=3375pq−1282p−1829q+500,(2)
HGO2G=43740pq−21960p−30196q+13328.Proof.
From the edge partition ofSi2C3IIIp,q given in Table 3, we have(1)
The first hyper-Gourava index forSi2C3IIIp,q is(13)HGO1G=∑uvϵEGdu+dv+du.dv2=1+3+1×321+2+2+2×222p+2q+2+3+2×328p+8q−14+3+3+3×3215pq−3p−13q+11=3375pq−1282p−1829q+500.(2)
The second hyper-Gourava index forSi2C3IIIp,q is(14)HGO2G=∑uvϵEGdu+dv.du.dv2=1+3×1×321+2+2×2×222p+2q+2+3×2×328p+8q−14+3+3×3×3215pq−13p−13q+11=43740pq−21960p−30196q+13328.


## 3.1.4. Gourava Indices for Silicon CarbideSiC3−IIIp,q

The unit cell ofSiC3−IIIp,q is shown in Figure 13. The 2D lattice graphs of SiC3−III5,1, SiC3−III5,2, and SiC3−III5,4 are shown in Figures 14–16, respectively. The edge partition of the edge set of SiC3−IIIp,q based on the degrees of end vertices is given in Table 4.Figure 13
Unit cell.Figure 14
SiC3−III5,1.Figure 15
SiC3−III5,2.Figure 16
SiC3−III5,4.Table 4
Degree-based edge partition ofSiC3IIIp,q.
du,dvFrequency1,221,312,23p+2q−32,36p+4q−83,312pq−12p−8q+8Theorem 7.
LetG be the graph of silicon carbide SiC3IIIp,q. Then, the first and second Gourava indices are(1)
GO1G=480pq−90p−60q+25,(2)
GO2G=648pq−420p−280q+168.Proof.
From the edge partition of the edge set ofSiC3IIIp,q given in Table 4, we have(1)
The first Gourava index forSiC3IIIp,q is(15)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×22+1+3+1×31+2+2+2×23p+2q−3+2+3+2×36p+4q−8+3+3+3×312pq−12p−8q+8=480pq−90p−60q+25.(2)
The second Gourava index forSiC3IIIp,q is(16)GO2G=∑uvϵEGdu+dv.du.dv=1+2×1×22+1+3×1×31+2+2×2×23p+2q−3+2+3×2×36p+4q−8+3+3×3×312pq−12p−8q+8=648pq−420p−280q+168.Theorem 8.
LetG be the graph of silicon carbide SiC3IIIp,q. Then, the first and second hyper-Gourava indices are(1)
HGO1G=2700pq−1782p−118q+739,(2)
HGO2G=34992pq−28824p−19216q+15576.Proof.
From the edge partition of the edge set ofSiC3IIIp,q given in Table 4, we have(1)
The first hyper-Gourava index forSiC3IIIp,q is(17)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×222+1+3+1×321+2+2+2×222p+2q+2+3+2×328p+8q−14+3+3+3×3215pq−3p−13q+11=2700pq−1782p−118q+739.(2)
The second hyper-Gourava index forSiC3IIIp,q is(18)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×222+1+3×1×321+2+2×2×222p+2q+2+3×2×328p+8q−14+3+3×3×3215pq−13p−13q+11=34992pq−28824p−19216q+15576.


## 3.1.5. Graphical Comparison of Results of Silicone Carbides

In Figures17–20, we can observe that the behavior of all indices is exponentially increasing with respect to the involved parameters.Figure 17
Comparison of the first Gourava index.Figure 18
Comparison of the second Gourava index.Figure 19
Comparison of the first hyper-Gourava index.Figure 20
Comparison of the second hyper-Gourava index.Codes for plotting the first and second Gourava indices for silicon carbideSi2C3Ip,r are given as follows:(19)plot3d225∗p∗q−61∗p−91∗q+18,p=0..1,q=0..1,colour=red,plot3d810∗p∗q−290∗p−430∗q+126,p=0..1,q=0..1,colour=green.


## 3.2. Gourava Indices for Bismuth Triiodide

BiI3 is an inorganic compound which is the result of the reaction between iodine and bismuth, which inspired the enthusiasm for subjective inorganic investigations [46]. BiI3 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 BiI3 gemstones are considered to be a three-layered stack structure in which a plane of bismuth atoms is sandwiched between iodide particle planes to form a continuous plane [49]. The periodic superposition of the three layers forms diamond-shaped BiI3 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 BiI3 has been integrated in [46]. We referred to [6] for the topological study of bismuth triiodide.


### 3.2.1. Bismuth Triiodide Chainm−BiI3

The molecular graph of the unit cell ofm−BiI3 is shown in Figure 21. From Figure 22, we can see that the molecular graph of m−BiI3 has two types of edge sets. The edge partition of the edge set of m−BiI3 is given in Table 5.Figure 21
Unit cell (bismuth triiodide).Figure 22
The chain form=3 (bismuth triiodide).Table 5
Degree-based edge partition ofm−Bil3m,n of end vertices of each edge.
du,dvFrequency1,64m+82,620m+4Theorem 9.
LetG be the graph of the bismuth triiodide chain m−BiI3. Then, the first and second Gourava indices are(1)
GO1G=452m+184,(2)
GO2G=2088m+720.Proof.
From the edge partition of the edge set ofm−BiI3 given in Table 5, we have(1)
The first Gourava index form−BiI3 is(20)GO1G=∑uvϵEGdu+dv+du.dv=1+6+1×64m+8+2+6+2×620m+4=452m+184.(2)
The second Gourava index form−BiI3 is(21)GO2G=∑uvϵEGdu+dv.du.dv=1+6×1×64m+8+2+6×2×620m+4=2088m+720.Theorem 10.
LetG be the graph of the bismuth triiodide chain m−BiI3. Then, the first and second hyper-Gourava indices are(1)
HGO1G=8676m+2952,(2)
HGO2G=191376m+93376.Proof.
From the edge partition of the edge set ofm−BiI3 given in Table 5, we have(1)
The first hyper-Gourava index form−BiI3 is(22)HGO1G=∑uvϵEGdu+dv+du.dv2=1+6+1×624m+8+2+6+2×6220m+4=8676m+2952.(2)
The second hyper-Gourava index form−BiI3 is(23)HGO2G=∑uvϵEGdu+dv.du.dv2=1+6×1×624m+8+2+6×2×6220m+4=191376m+93376.


### 3.2.2. Bismuth Triiodide SheetBiI3m×n

The molecular graph of the bismuth triiodide sheetBiI3m×n is shown in Figure 23. It can be observed from Figure 23 that the edge set of the bismuth triiodide sheet BiI3m×n can be divided into three classes based on the degrees of end vertices as shown in Table 6.Figure 23
Bismuth triiodide sheet.Table 6
Degree-based edge partition ofBiI3m×n.
du,dvFrequency1,64m+n+12,643mn+2m+2n−13,66nm−1Theorem 11.
LetG be the graph of the bismuth triiodide sheet BiI3m×n. Then, the first and second Gourava indices are(1)
GO1G=402mn+212m+50n−28,(2)
GO2G=2124mn+936m−36n−216.Proof.
From the edge partition of the edge set of the bismuth triiodide sheetBiI3m×n given in Table 6, we have(1)
The first Gourava index forBiI3m×n is(24)GO1G=∑uvϵEGdu+dv+du.dv=1+6+1×64m+4n+4+2+6+2×612mn+8m+8n−4+3+6+3×66mn−6n=402mn+212m+50n−28.(2)
The second Gourava index forBiI3m×n is(25)GO2G=∑uvϵEGdu+dv.du.dv=1+6×1×64m+4n+4+2+6×2×612mn+8m+8n−4+3+6×3×66mn−6n=2124mn+936m−36n−216.Theorem 12.
LetG be the graph of the bismuth triiodide sheet BiI3m×n. Then, the first and second hyperGourava indices are(1)
HGO1G=9174mn+3876m−498n−924,(2)
HGO2G=268056mn+80784m−143036n−29808.Proof.
From the edge partition of the edge set of the bismuth triiodide sheetBiI3m×n given in Table 6, we have(1)
The first hyper-Gourava index forBiI3m×n is(26)HGO1G=∑uvϵEGdu+dv+du.dv2=1+6+1×624m+4n+4+2+6+2×6212mn+8m+8n−4+3+6+3×626mn−6n=9174mn+3876m−498n−924.(2)
The second hyper-Gourava index forBiI3m×n is(27)HGO2G=∑uvϵEGdu+dv.du.dv2=1+6×1×624m+4n+4+2+6×2×6212mn+8m+8n−4+3+6×3×626mn−6n=268056mn+80784m−143036n−29808.


## 3.2.1. Bismuth Triiodide Chainm−BiI3

The molecular graph of the unit cell ofm−BiI3 is shown in Figure 21. From Figure 22, we can see that the molecular graph of m−BiI3 has two types of edge sets. The edge partition of the edge set of m−BiI3 is given in Table 5.Figure 21
Unit cell (bismuth triiodide).Figure 22
The chain form=3 (bismuth triiodide).Table 5
Degree-based edge partition ofm−Bil3m,n of end vertices of each edge.
du,dvFrequency1,64m+82,620m+4Theorem 9.
LetG be the graph of the bismuth triiodide chain m−BiI3. Then, the first and second Gourava indices are(1)
GO1G=452m+184,(2)
GO2G=2088m+720.Proof.
From the edge partition of the edge set ofm−BiI3 given in Table 5, we have(1)
The first Gourava index form−BiI3 is(20)GO1G=∑uvϵEGdu+dv+du.dv=1+6+1×64m+8+2+6+2×620m+4=452m+184.(2)
The second Gourava index form−BiI3 is(21)GO2G=∑uvϵEGdu+dv.du.dv=1+6×1×64m+8+2+6×2×620m+4=2088m+720.Theorem 10.
LetG be the graph of the bismuth triiodide chain m−BiI3. Then, the first and second hyper-Gourava indices are(1)
HGO1G=8676m+2952,(2)
HGO2G=191376m+93376.Proof.
From the edge partition of the edge set ofm−BiI3 given in Table 5, we have(1)
The first hyper-Gourava index form−BiI3 is(22)HGO1G=∑uvϵEGdu+dv+du.dv2=1+6+1×624m+8+2+6+2×6220m+4=8676m+2952.(2)
The second hyper-Gourava index form−BiI3 is(23)HGO2G=∑uvϵEGdu+dv.du.dv2=1+6×1×624m+8+2+6×2×6220m+4=191376m+93376.


## 3.2.2. Bismuth Triiodide SheetBiI3m×n

The molecular graph of the bismuth triiodide sheetBiI3m×n is shown in Figure 23. It can be observed from Figure 23 that the edge set of the bismuth triiodide sheet BiI3m×n can be divided into three classes based on the degrees of end vertices as shown in Table 6.Figure 23
Bismuth triiodide sheet.Table 6
Degree-based edge partition ofBiI3m×n.
du,dvFrequency1,64m+n+12,643mn+2m+2n−13,66nm−1Theorem 11.
LetG be the graph of the bismuth triiodide sheet BiI3m×n. Then, the first and second Gourava indices are(1)
GO1G=402mn+212m+50n−28,(2)
GO2G=2124mn+936m−36n−216.Proof.
From the edge partition of the edge set of the bismuth triiodide sheetBiI3m×n given in Table 6, we have(1)
The first Gourava index forBiI3m×n is(24)GO1G=∑uvϵEGdu+dv+du.dv=1+6+1×64m+4n+4+2+6+2×612mn+8m+8n−4+3+6+3×66mn−6n=402mn+212m+50n−28.(2)
The second Gourava index forBiI3m×n is(25)GO2G=∑uvϵEGdu+dv.du.dv=1+6×1×64m+4n+4+2+6×2×612mn+8m+8n−4+3+6×3×66mn−6n=2124mn+936m−36n−216.Theorem 12.
LetG be the graph of the bismuth triiodide sheet BiI3m×n. Then, the first and second hyperGourava indices are(1)
HGO1G=9174mn+3876m−498n−924,(2)
HGO2G=268056mn+80784m−143036n−29808.Proof.
From the edge partition of the edge set of the bismuth triiodide sheetBiI3m×n given in Table 6, we have(1)
The first hyper-Gourava index forBiI3m×n is(26)HGO1G=∑uvϵEGdu+dv+du.dv2=1+6+1×624m+4n+4+2+6+2×6212mn+8m+8n−4+3+6+3×626mn−6n=9174mn+3876m−498n−924.(2)
The second hyper-Gourava index forBiI3m×n is(27)HGO2G=∑uvϵEGdu+dv.du.dv2=1+6×1×624m+4n+4+2+6×2×6212mn+8m+8n−4+3+6×3×626mn−6n=268056mn+80784m−143036n−29808.


## 3.3. Graphical Representation

Graphical representation of computed topological indices for the bismuth triiodide chain is shown in Figures24–27, and the graphical representation of the bismuth triiodide sheet is shown in Figures 28–31.Figure 24
Plots of the first Gourava index form−Bil3.Figure 25
Plots of the second Gourava index form−Bil3.Figure 26
Plots of the first hyper-Gourava index form−Bil3.Figure 27
Plots of the second hyper-Gourava index form−Bil3.Figure 28
Plots of the first Gourava index forBil3m×n.Figure 29
Plots of the second Gourava index forBil3m×n.Figure 30
Plots of the first hyper-Gourava index forBil3m×n.Figure 31
Plots of the second hyper-Gourava index forBil3m×n.


## 3.4. Gourava Indices for Dendrimers

In the medication mathematical model, the structure of the drug is addressed as an undirected graph, where each vertex exhibits a molecule and each edge addresses a bond between atoms. A huge number of new drugs have been made each year. From this time forward, it asks for a giant measure of work to choose the pharmacological compound and organic qualities of these new drugs, and such remaining tasks at hand end up being progressively specific and grouped. It requires enough reagent rigging and accomplices to test the exhibitions and the responses of new drugs. Nevertheless, in cut down poor countries and locales (for instance, certain urban networks and countries in South America, Southeast Asia, Africa, and India), there is no sufficient money to settle reagents and apparatus which can be used to gauge the biochemical properties. For topological study of dendrimers, we refer [53–66].


### 3.4.1. Gourava Indices of Porphyrin DendrimerDnPn

The algebraic graph of porphyrin dendrimerDnPn is shown in Figure 32. For porphyrin dendrimer DnPn, VDnPn=96n−10 and EDnPn=105n−11. There 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 DnPn is given in Table 7.Figure 32
Porphyrin dendrimerDnPn.Table 7
Degree-based edge partition ofDnPn.
du,dvFrequency1,32n1,4124n2,210n−52,348n−63,313n3,48nTheorem 13.
LetG be the graph of porphyrin dendrimer DnPn. Then, the first and second Gourava indices are(1)
GO1G=1169n−106,(2)
GO2G=3478n−260.Proof.
From the edge partition ofDnPn given in Table 7, we have(1)
The first Gourava index forDnPn is(28)GO1G=∑uvϵEGdu+dv+du.dv=1+3+1×32n+1+4+1×424n+2+2+2×210n−5+2+3+2×348n−6+3+3+3×313n+3+4+3×48n=1169n−106.(2)
The second Gourava index forDnPn is(29)GO2DnPn=∑uvϵEGdu+dv.du.dv=1+3×1×32n+1+4×1×424n+2+2×2×210n−5+2+3×2×348n−6+3+3×3×313n+3+4×3×48n=3478n−260.Theorem 14.
LetG be the graph of porphyrin dendrimer DnPn. Then, the first and second hyper-Gourava indices are(1)
HGO1G=13727n−1046,(2)
HGO2G=150004n−6680.Proof.
From the edge partition ofDnPn given in Table 7, we have(1)
The first hyper-Gourava index forDnPn is(30)HGO1G=∑uvϵEGdu+dv+du.dv2=1+3+1×322n+1+4+1×4224n+2+2+2×2210n−5+2+3+2×3248n−6+3+3+3×3213n+3+4+3×428n=13727n−1046.(2)
The second hyper-Gourava index forDnPn is(31)HGO2G=∑uvϵEGdu+dv.du.dv2=1+3×1×322n+1+4×1×4224n+2+2×2×2210n−5+2+3×2×3248n−6+3+3×3×3213n+3+4×3×48n=150004n−6680.


### 3.4.2. Propyl Ether Imine Dendrimer (PETIM)

The algebraic graph of propyl ether imine dendrimer (PETIM) is shown in Figure 33. For propyl ether imine dendrimer (PETIM), VPETIM=24.2n−23 and EPETIM=24.2n−24. There 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 propyl ether imine dendrimer (PETIM) is given in Table 8.Figure 33
Propyl ether imine dendrimer (PETIM).Table 8
Degree-based edge partition of (PETIM).
du,dvFrequency1,22n+12,22n+4−182,36.2n−6Theorem 15.
LetG be the graph of propyl ether imine dendrimer (PETIM). Then, the first and second Gourava indices are(1)
GO1G=66.2n+5nn+1+8.2n+4−232,(2)
GO2G=18.2n+6.nn+1+16.2n+4−528.Proof.
From the edge partition ofPETIM given in Table 8, we have(1)
The first Gourava index forPETIM is(32)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×22n+1+2+2+2×22n+4−18+2+3+2×36.2n−6=66.2n+5nn+1+8.2n+4−232.(2)
The second Gourava index forPETIM is(33)GO2PETIM=∑uvϵEGdu+dv.du.dv=1+2×1×22n+1+2+2×2×22n+4−18+2+3×2×36.2n−6=18.2n+6.nn+1+16.2n+4−528.Theorem 16.
LetG be the graph of propyl ether imine dendrimer (PETIM). Then, the first and second hyper-Gourava indices are(1)
HGO1G=726.2n+25.nn+1+64.2n+4−2120,(2)
HGO2G=5400.2n+36.nn+1+256.2n+4−11808.Proof.
From the edge partition ofPETIM given in Table 8, we have(1)
The first hyper-Gourava index forPETIM is(34)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×222n+1+2+2+2×222n+4−18+2+3+2×326.2n−6=726.2n+25.nn+1+64.2n+4−2120.(2)
The second hyper-Gourava index forPETIM is(35)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×222n+1+2+2×2×222n+4−18+2+3×2×36.2n−6=5400.2n+36.nn+1+256.2n+4−11808.


### 3.4.3. Zinc-Porphyrin DendrimerDPZn

The algebraic graph of zinc-porphyrin dendrimerDPZn is shown in Figure 34. There 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 DPZn is given in Table 9.Figure 34
Zinc-porphyrin dendrimerDPZn.Table 9
Degree-based edge partition ofDPZn.
du,dvFrequency2,216.2n−42,340.2n−163,38.2n−163,44Theorem 17.
LetG be the graph of zinc-porphyrin dendrimer DPZn. Then, the first and second Gourava indices are(1)
GO1G=688.2n−372,(2)
GO2G=1888.2n−1072.Proof.
From the edge partition ofDPZn given in Table 9, we have(1)
The first Gourava index forDPZn is(36)GO1G=∑uvϵEGdu+dv+du.dv=2+2+2×216.2n−4+2+3+2×340.2n−16+3+3+3×38.2n−16+3+4+3×44=688.2n−372.(2)
The second Gourava index forDPZn is(37)GO2G=∑uvϵEGdu+dv.du.dv=2+2×2×216.2n−4+2+3×2×340.2n−16+3+3×3×38.2n−16+3+4×3×44=1888.2n−1072.Theorem 18.
LetG be the graph of zinc-porphyrin dendrimer DPZn. Then, the first and second hyper-Gourava indices are(1)
HGO1G=7664nn−5716,(2)
HGO2G=63424.2n−33856.Proof.
From the edge partition ofDPZn given in Table 9, we have(1)
The first hyper-Gourava index forDPZn is(38)HGO1G=∑uvϵEGdu+dv+du.dv2=2+2+2×2216.2n−4+2+3+2×3240.2n−16+3+3+3×328.2n−16+3+4+3×424=7664nn−5716.(2)
The second hyper-Gourava index forDPZn is(39)HGO2G=∑uvϵEGdu+dv.du.dv2=2+2×2×2216.2n−4+2+3×2×3240.2n−16+3+3×3×328.2n−16+3+4×3×424=63424.2n−33856.


### 3.4.4. Poly(EThyleneAmidoAmine) Dendrimer (PETAA)

The algebraic graph of Poly(EThyleneAmidoAmine) dendrimer (PETAA) is shown in Figure 35. For Poly(EThyleneAmidoAmine) dendrimer (PETAA), VPETAA=44.2n−18 and EPETAA=44.2n−19. There 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 Poly(EThyleneAmidoAmine) dendrimer (PETAA) is given in Table 10.Figure 35
Poly(EThylene amide amine) dendrimer (PETAA).Table 10
Degree-based edge partition of (PETAA).
du,dvFrequency1,24.2n1,34.2n−22,216.2n−82,320.2n−9Theorem 19.
LetG be the graph of Poly(EThylene Amide Amine) dendrimer (PETAA). Then, the first and second Gourava indices are(1)
GO1G=369.2n−177,(2)
GO2G=928.2n−422.Proof.
From the edge partition ofPETAA given in Table 10, we have(1)
The first Gourava index forPETAA is(40)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×24.2n+1+3+1×34.2n−2+2+2+2×216.2n−8+2+3+2×320.2n−9=369.2n−177.(2)
The second Gourava index forPETAA is(41)GO2PETAA=∑uvϵEGdu+dv.du.dv=1+2×1×24.2n+1+3×1×34.2n−2+2+2×2×216.2n−8+2+3×2×320.2n−9=928.2n−422.Theorem 20.
LetG be the graph of Poly(EThylene Amide Amine) dendrimer (PETAA). Then, the first and second hyper-Gourava indices are(1)
HGO1G=3740.2n−1699,(2)
HGO2G=22816.2n−10436.Proof.
From the edge partition ofPETAA given in Table 10, we have(1)
The first hyper-Gourava index forPETAA is(42)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×224.2n+1+3+1×324.2n−2+2+2+2×2216.2n−8+2+3+2×3220.2n−9=3740.2n−1699.(2)
The second hyper-Gourava index forPETAA is(43)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×224.2n+1+3×1×324.2n−2+2+2×2×2216.2n−8+2+3×2×3220.2n−9=22816.2n−10436.


### 3.4.5. Graphical Comparison

In this section, we will present the graphical comparison of first, second, first hyper-, and second hyper-Gourava indices for porphyrin dendrimerDnPn, propyl ether imine dendrimer (PETIM), zinc-porphyrin dendrimer DPZn and Poly(EThyleneAmidoAmine) dendrimer (PETAA). Figures 36–39 show the all indices are linearly increasing with respect to involved parameters.Figure 36
First Gourava index.Figure 37
Second Gourava index.Figure 38
First hyper-Gourava index.Figure 39
Second hyper-Gourava index.


## 3.4.1. Gourava Indices of Porphyrin DendrimerDnPn

The algebraic graph of porphyrin dendrimerDnPn is shown in Figure 32. For porphyrin dendrimer DnPn, VDnPn=96n−10 and EDnPn=105n−11. There 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 DnPn is given in Table 7.Figure 32
Porphyrin dendrimerDnPn.Table 7
Degree-based edge partition ofDnPn.
du,dvFrequency1,32n1,4124n2,210n−52,348n−63,313n3,48nTheorem 13.
LetG be the graph of porphyrin dendrimer DnPn. Then, the first and second Gourava indices are(1)
GO1G=1169n−106,(2)
GO2G=3478n−260.Proof.
From the edge partition ofDnPn given in Table 7, we have(1)
The first Gourava index forDnPn is(28)GO1G=∑uvϵEGdu+dv+du.dv=1+3+1×32n+1+4+1×424n+2+2+2×210n−5+2+3+2×348n−6+3+3+3×313n+3+4+3×48n=1169n−106.(2)
The second Gourava index forDnPn is(29)GO2DnPn=∑uvϵEGdu+dv.du.dv=1+3×1×32n+1+4×1×424n+2+2×2×210n−5+2+3×2×348n−6+3+3×3×313n+3+4×3×48n=3478n−260.Theorem 14.
LetG be the graph of porphyrin dendrimer DnPn. Then, the first and second hyper-Gourava indices are(1)
HGO1G=13727n−1046,(2)
HGO2G=150004n−6680.Proof.
From the edge partition ofDnPn given in Table 7, we have(1)
The first hyper-Gourava index forDnPn is(30)HGO1G=∑uvϵEGdu+dv+du.dv2=1+3+1×322n+1+4+1×4224n+2+2+2×2210n−5+2+3+2×3248n−6+3+3+3×3213n+3+4+3×428n=13727n−1046.(2)
The second hyper-Gourava index forDnPn is(31)HGO2G=∑uvϵEGdu+dv.du.dv2=1+3×1×322n+1+4×1×4224n+2+2×2×2210n−5+2+3×2×3248n−6+3+3×3×3213n+3+4×3×48n=150004n−6680.


## 3.4.2. Propyl Ether Imine Dendrimer (PETIM)

The algebraic graph of propyl ether imine dendrimer (PETIM) is shown in Figure 33. For propyl ether imine dendrimer (PETIM), VPETIM=24.2n−23 and EPETIM=24.2n−24. There 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 propyl ether imine dendrimer (PETIM) is given in Table 8.Figure 33
Propyl ether imine dendrimer (PETIM).Table 8
Degree-based edge partition of (PETIM).
du,dvFrequency1,22n+12,22n+4−182,36.2n−6Theorem 15.
LetG be the graph of propyl ether imine dendrimer (PETIM). Then, the first and second Gourava indices are(1)
GO1G=66.2n+5nn+1+8.2n+4−232,(2)
GO2G=18.2n+6.nn+1+16.2n+4−528.Proof.
From the edge partition ofPETIM given in Table 8, we have(1)
The first Gourava index forPETIM is(32)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×22n+1+2+2+2×22n+4−18+2+3+2×36.2n−6=66.2n+5nn+1+8.2n+4−232.(2)
The second Gourava index forPETIM is(33)GO2PETIM=∑uvϵEGdu+dv.du.dv=1+2×1×22n+1+2+2×2×22n+4−18+2+3×2×36.2n−6=18.2n+6.nn+1+16.2n+4−528.Theorem 16.
LetG be the graph of propyl ether imine dendrimer (PETIM). Then, the first and second hyper-Gourava indices are(1)
HGO1G=726.2n+25.nn+1+64.2n+4−2120,(2)
HGO2G=5400.2n+36.nn+1+256.2n+4−11808.Proof.
From the edge partition ofPETIM given in Table 8, we have(1)
The first hyper-Gourava index forPETIM is(34)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×222n+1+2+2+2×222n+4−18+2+3+2×326.2n−6=726.2n+25.nn+1+64.2n+4−2120.(2)
The second hyper-Gourava index forPETIM is(35)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×222n+1+2+2×2×222n+4−18+2+3×2×36.2n−6=5400.2n+36.nn+1+256.2n+4−11808.


## 3.4.3. Zinc-Porphyrin DendrimerDPZn

The algebraic graph of zinc-porphyrin dendrimerDPZn is shown in Figure 34. There 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 DPZn is given in Table 9.Figure 34
Zinc-porphyrin dendrimerDPZn.Table 9
Degree-based edge partition ofDPZn.
du,dvFrequency2,216.2n−42,340.2n−163,38.2n−163,44Theorem 17.
LetG be the graph of zinc-porphyrin dendrimer DPZn. Then, the first and second Gourava indices are(1)
GO1G=688.2n−372,(2)
GO2G=1888.2n−1072.Proof.
From the edge partition ofDPZn given in Table 9, we have(1)
The first Gourava index forDPZn is(36)GO1G=∑uvϵEGdu+dv+du.dv=2+2+2×216.2n−4+2+3+2×340.2n−16+3+3+3×38.2n−16+3+4+3×44=688.2n−372.(2)
The second Gourava index forDPZn is(37)GO2G=∑uvϵEGdu+dv.du.dv=2+2×2×216.2n−4+2+3×2×340.2n−16+3+3×3×38.2n−16+3+4×3×44=1888.2n−1072.Theorem 18.
LetG be the graph of zinc-porphyrin dendrimer DPZn. Then, the first and second hyper-Gourava indices are(1)
HGO1G=7664nn−5716,(2)
HGO2G=63424.2n−33856.Proof.
From the edge partition ofDPZn given in Table 9, we have(1)
The first hyper-Gourava index forDPZn is(38)HGO1G=∑uvϵEGdu+dv+du.dv2=2+2+2×2216.2n−4+2+3+2×3240.2n−16+3+3+3×328.2n−16+3+4+3×424=7664nn−5716.(2)
The second hyper-Gourava index forDPZn is(39)HGO2G=∑uvϵEGdu+dv.du.dv2=2+2×2×2216.2n−4+2+3×2×3240.2n−16+3+3×3×328.2n−16+3+4×3×424=63424.2n−33856.


## 3.4.4. Poly(EThyleneAmidoAmine) Dendrimer (PETAA)

The algebraic graph of Poly(EThyleneAmidoAmine) dendrimer (PETAA) is shown in Figure 35. For Poly(EThyleneAmidoAmine) dendrimer (PETAA), VPETAA=44.2n−18 and EPETAA=44.2n−19. There 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 Poly(EThyleneAmidoAmine) dendrimer (PETAA) is given in Table 10.Figure 35
Poly(EThylene amide amine) dendrimer (PETAA).Table 10
Degree-based edge partition of (PETAA).
du,dvFrequency1,24.2n1,34.2n−22,216.2n−82,320.2n−9Theorem 19.
LetG be the graph of Poly(EThylene Amide Amine) dendrimer (PETAA). Then, the first and second Gourava indices are(1)
GO1G=369.2n−177,(2)
GO2G=928.2n−422.Proof.
From the edge partition ofPETAA given in Table 10, we have(1)
The first Gourava index forPETAA is(40)GO1G=∑uvϵEGdu+dv+du.dv=1+2+1×24.2n+1+3+1×34.2n−2+2+2+2×216.2n−8+2+3+2×320.2n−9=369.2n−177.(2)
The second Gourava index forPETAA is(41)GO2PETAA=∑uvϵEGdu+dv.du.dv=1+2×1×24.2n+1+3×1×34.2n−2+2+2×2×216.2n−8+2+3×2×320.2n−9=928.2n−422.Theorem 20.
LetG be the graph of Poly(EThylene Amide Amine) dendrimer (PETAA). Then, the first and second hyper-Gourava indices are(1)
HGO1G=3740.2n−1699,(2)
HGO2G=22816.2n−10436.Proof.
From the edge partition ofPETAA given in Table 10, we have(1)
The first hyper-Gourava index forPETAA is(42)HGO1G=∑uvϵEGdu+dv+du.dv2=1+2+1×224.2n+1+3+1×324.2n−2+2+2+2×2216.2n−8+2+3+2×3220.2n−9=3740.2n−1699.(2)
The second hyper-Gourava index forPETAA is(43)HGO2G=∑uvϵEGdu+dv.du.dv2=1+2×1×224.2n+1+3×1×324.2n−2+2+2×2×2216.2n−8+2+3×2×3220.2n−9=22816.2n−10436.


## 3.4.5. Graphical Comparison

In this section, we will present the graphical comparison of first, second, first hyper-, and second hyper-Gourava indices for porphyrin dendrimerDnPn, propyl ether imine dendrimer (PETIM), zinc-porphyrin dendrimer DPZn and Poly(EThyleneAmidoAmine) dendrimer (PETAA). Figures 36–39 show the all indices are linearly increasing with respect to involved parameters.Figure 36
First Gourava index.Figure 37
Second Gourava index.Figure 38
First hyper-Gourava index.Figure 39
Second hyper-Gourava index.


## 4. 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. There are more than around 148 topological indices, but none of them can completely describe all properties of a chemical compound. Therefore, 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.



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