# Analysis of Bismuth (III) Iodide and Dendrimers in Drug Applications

Authors: Shahid Amin; M. A. Rehman; Amir Naseem; Jihad Younis; Syed Sheraz Asghar
Journal: Journal of Chemistry
 (2022)

---

## Abstract

Dendrimers are spherical three-dimensional molecules with a repetitively branching core. They are normally symmetric around the core. Bismuth (III) iodide has the formulaBiI3 and is an inorganic chemical. The reaction between bismuth and iodine produces this gray-black solid, which was of great interest in qualitative inorganic analysis. Mathematical chemistry is an area of mathematics that employs mathematical methods to tackle chemical-related problems. One of these tools is a graphical representation of chemical molecules, known as the molecular graph of a chemical substance. A topological index (TI) is a mathematical function that assigns a numerical value to a (molecular) graph and predicts many physical, chemical, biological, thermodynamical, and structural features of that network. In this work, we will calculate a new topological index, namely, Sombor index, multiplicative Sombor index, and its reduced version for bismuth (III) iodide and dendrimers. We also plot our computed results of Sombor index, multiplicative Sombor index, and reduced Sombor index to examine how they were affected by the parameters involved.


---

## Body

## 1. Introduction

LetG be a simply connected graph with the vertex and edge sets V (G) and E (G) of order n. Let i∈VG and its degree is represented by di. The ordered pair (i, j) is an edge eij of G if we have i,j∈EG. The total number of vertices is called order of G, while total number of edges is called size of the graph. Chemical graphs are graphs in which the vertices and edges represent the atoms and bonds of a chemical molecule. The vertices and edges are identified according to the atoms and types of bonds they represent [1, 2].Topological invariants are numeric values which provide us the knowledge about a chemical structure as well as reveal its underlying qualities while having to do trials [3, 4]. Distance-based, degree-based, and eccentricity-based topological indices are the three major topological indices. Degree-based topological indices are particularly important among these groups and are useful tools for chemists [5–7]. Distance-based topological indices are concerned with graph distances. Degree-based topological indices are concerned with the concept of degree and eccentricity-based topological indices are concerned with the eccentricity of vertices [8, 9]. Topological indices have broad uses in material science, math, informatics, biology, and other fields [10–12]. Furthermore, it is used in nonexact quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR). Chemical structure and TIs are interconnected [13, 14]. Physical and chemical parameters of the underlying molecule can be determined from TIs, which include heat of formation, surface tension, heat of evaporation, chromatographic retention, vapour pressure, and boiling point [15, 16].The Wiener index was the first and most widely researched topological index [17]. In chemistry, it was the first molecular topological index to be employed [18, 19]. The boiling temperatures of alkane molecules are closely associated with the Wiener index number, according to Wiener [20–22]. Later research into quantitative structure-activity relationships revealed that it is also correlated with other variables such as the critical point parameters, the density, surface tension and viscosity of the liquid phase, and the molecule’s Van der Waals surface area [23, 24]. Randic index is a simple topological index which is introduced by Milan randic [25, 26]. Zagreb indices is the oldest topological indices written by Gutman and Trinajstic [27]. In QSPRs, topological indices are utilised to guess the properties of the concerned compound. There is no such type of topological index which give us an idea of all the properties of the concerned compound. As a result, there is constantly a need for new topological indices to be defined. Sombor indices and reduced Sombor indices have lately been characterised as(1)SOG=∑ij∈EGdi2+dj2,(2)SOredG=∑ij∈EGdi−12+dj−12.The multiplicative Sombor index and multiplicative reduced Sombor index of a graphG are defined as(3)SOIIG=∏ij∈EGdi2+dj2,(4)SOIIredG=∏ij∈EGdi−12+dj−12.We compute Sombor indices, multiplicative Sombor indices, and reduced Sombor indices for bismuth (III) iodide and dendrimers as well as their graphical representations, in this study [28–30].


## 2. Methodology

We first built a graph of the molecular compounds and counted the total number of vertices and edges to arrive at our conclusions. Second, based on the degrees of the end vertices, we classified the graphs edge set into several types. We arrived at our desired findings by using Sombor indices definitions. We plot our computed findings to examine how they were affected by the parameters involved.


## 3. Results

The main computational results are presented in this section. We present results about bismuth (III) iodide and dendrimers.


### 3.1. Bismuth Triiodide

m−BiI3 is an inorganic chemical formed by the interaction of iodine and bismuth. It is this compound that sparked interest in subjective inorganic research [20]. m−BiI3 is an excellent inorganic chemical that comes in qualitative inorganic analysis [31]. Bi-adoped glass optical strands have been shown to be one of the most promising dynamic laser media. Bi-adoped fibre lasers and optical loudspeakers have been built using several types of bi-adoped fibre strands [32]. Layered m−BiI3 gemstones have a three-layered stack structure, with a bismuth atom plane sandwiched between iodide particle planes to form a continuous plane. Diamond-shaped m−BiI3 crystals with R-3 symmetry are formed by the periodic superposition of three layers. A hexagonal structure with symmetry is formed by a progressive stack of I-Bi-I layers [8, 33].


### 3.2. Bismuth Triiodide Chainm−BiI3

The molecular graph of the unit cell ofm−BiI3 is shown in Figure 1. From Figure 2, 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 1. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 3.Figure 1
Unit cell (bismuth triiodide).Figure 2
The chain form = 3 (bismuth triiodide).Table 1
Edge partitioning based on degree ofm−BiI3m,n.
EEdi,djFrequencyE1E1,64m+8E2E2,620m+4Figure 3
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 1.
TheSO and SOred indices for m−BiI3 are as follows:(1)
SOG=4m+837+220m+410(2)
SOredG=40+20m+45m+126Proof.
From the edge partitioning based on degree ofm−BiI3, we have following computations for SO and SOred indices:(5)SOG=∑ij∈EGdi2+dj2=12+624m+8+22+6220m+4=4m+837+220m+410,(6)SOredG=∑ij∈EGdi−12+dj−12=1−12+6−124m+8+2−12+6−1220m+4=40+20m+45m+126.Theorem 2.
TheSOII and SOIIred indices for m−BiI3 are as follows:(1)
SOIIG=280m2+176m+32370(2)
SOIIredG=320m5m+126Proof.
From the edge partitioning based on degree ofm−BiI3, we have following computations for SOII and SOIIred indices:(7)SOIIG=∏ij∈EGdi2+dj2=12+624m+8×22+6220m+4=280m2+176m+32370,(8)SOIIredG=∏ij∈EGdi−12+dj−12=1−12+6−124m+8×2−12+6−1220m+4=320m5m+126.


### 3.3. Bismuth Triiodide SheetBiI3m×n

The molecular graph of the bismuth triiodide sheetBiI3m×n is shown in Figure 4. It can be observed from Figure 3 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 given in Table 2. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 5.Figure 4
Bismuth triiodide sheet.Table 2
Edge partitioning based on degree ofm−BiI3m×n.
EEdi,djFrequencyE1E1,64m+n+1E2E2,643mn+2m+2n−1E3E3,66nm−1Figure 5
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 3.
TheSO and SOred indices for m−BiI3m×n are as follows:(1)
SOG=4m+n+137+83mn+2m+2n−110+18nm−15(2)
SOredG=20m+n+1+43mn+2m+2n−126+6nm−129Proof.
From the edge partitioning based on degree ofm−BiI3m×n, we have following computations for SO and SOred indices:(9)SOG=∑ij∈EGdi2+dj2=412+62m+n+1+422+623mn+2m+2n−1+32+626nm−1=4m+n+137+83mn+2m+2n−110+18nm−15,(10)SOredG=∑ij∈EGdi−12+dj−12=41−12+6−12m+n+1+42−12+6−123mn+2m+2n−1+6nm−13−12+6−12=20m+n+1+43mn+2m+2n−126+6nm−129.Theorem 4.
TheSOII and SOIIred indices for m−BiI3m×n are as follows:(1)
SOIIG=28803m3n+2m3+4m2n−m2+3m2n2−4mn2−6mn−2m−2n2−n+174(2)
SOIIredG=4803m3n+2m3+4m2n−m2+3m2n2−3mn2−6mn−2m−2n2−n+1754Proof.
From the edge partitioning based on degree ofm−BiI3m×n, we have following computations for SOII and SOIIred indices:(11)SOIIG=∑ij∈EGdi2+dj2=412+62m+n+1×422+623mn+2m+2n−1×6nm−132+62)=28803m3n+2m3+4m2n−m2+3m2n2−4mn2−6mn−2m−2n2−n+174,(12)SOIIredG=∑ij∈EGdi−12+dj−12=41−12+6−12m+n+1×42−12+6−123mn+2m+2n−1×6nm−13−12+6−12=4803m3n+2m3+4m2n−m2+3m2n2−3mn2−6mn−2m−2n2−n+1754.


### 3.4. Dendrimers

The structure of medicine is represented mathematically as an undirected graph, with each vertex representing a molecule and each edge representing an atom-atom connection [34]. Every year, a large number of new medications are developed. Furthermore, selecting the pharmacological substance and organic properties of these new medications necessitates a significant amount of effort and subsequent duties become increasingly detailed and grouped. To test the exhibitions and the responses of new medications, enough reagent rigs and accomplices are required [35, 36]. Nonetheless, in impoverished countries and communities (for example, particular metropolitan networks and countries in South America, Southeast Asia, Africa, and India), there is insufficient funding to purchase reagents and equipment that can be used to assess biochemical attributes. Because of their effects against prion disorders, Alzheimer’s disease, inflammation, HIV, herpes simplex virus (HSV), bacteria, and cancer, dendrimers are interested in medication research. We use the term “topological research of dendrimers” to refer to the study of dendrimers on a topological level [29, 30].


### 3.5. Porphyrin DendrimerDnPn

The algebraic graph ofDnPn is shown in Figure 6. For DnPn, VDnPn=96n−10 and EDnPn=105n−11. Based on the degree of end vertices, there are six types of edges in the edge set of DnPn. Degree-based partition of edges of DnPn is given in Table 3. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 7.Figure 6
Porphyrin dendrimerDnPn.Table 3
Edge partitioning based on degree ofDnPn.
EEdi,djFrequencyE1E1,32nE2E1,4124nE3E2,210n−5E4E2,348n−6E5E3,313nE6E3,48nFigure 7
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 5.
TheSO and SOred indices for DnPn are as follows:(1)
SOG=40n+2n10+12417+59n−102+48n−613s(2)
SOredG=376n+36n−52+48n−65+8n13Proof.
From the edge partitioning based on degree ofDnPn, we have following computations for SO and SOred indices:(13)SOG=∑ij∈EGdi2+dj2=12+322n+12+42124n+22+2210n−5+22+3248n−6+32+3213n+32+428n=40n+2n10+12417+59n−102+48n−613,(14)SOredG=∑ij∈EGdi−12+dj−12=1−12+3−122n+1−12+4−12124n+2−12+2−1210n−5+2−12+3−1248n−6+3−12+3−1213n+3−12+4−128n=376n+36n−52+48n−65+8n13.Theorem 6.
TheSOII and SOIIred indices for DnPn are as follows:(1)
SOIIG=408n2832n2−834n+601105(2)
SOIIredG=1408n1728n2−456n+30165Proof.
From the edge partitioning based on degree ofDnPn, we have following computations for SOII and SOIIred indices:(15)SOIIG=∑ij∈EGdi2+dj2=12+322n×12+42124n×22+2210n−5×22+3248n−6×32+3213n×32+428n=408n2832n2−834n+601105,(16)SOIIredG=∑ij∈EGdi−12+dj−12=1−12+3−122n×1−12+4−12124n×2−12+2−1210n−5×2−12+3−1248n−6×3−12+3−1213n×3−12+4−128n=1408n1728n2−456n+30165.


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

The algebraic graph of PETIM is shown in Figure8. For PETIM, VPETIM=242n−23 and EPETIM=242n−24. There are six type of edges in the edge set of PETIM, based on the degree of end vertices. Degree-based partition of edges of PETIM is given in Table 4. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 9.Figure 8
Propyl ether imine (PETIM) dendrimer.Table 4
Edge partitioning based on degree of PETIM.
EEdi,djFrequencyE1E1,22n+1E2E2,22n+4−18E3E2,362n−6Figure 9
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 7.
TheSO and SOred indices for PETIM are as follows:(1)
SOG=2n+15+42n+3−92+62n−113(2)
SOredG=2n+1+22n+3−92+62n−15Proof.
From the edge partitioning based on degree of PETIM, we have following computations forSO and SOred indices:(17)SOG=∑ij∈EGdi2+dj2=12+222n+1+22+222n+4−18+22+3262n−6=2n+15+42n+3−92+62n−113,(18)SOredG=∑ij∈EGdi−12+dj−12=1−12+2−122n+1+2−12+2−122n+4−18+2−12+3−1262n−6=2n+1+22n+3−92+62n−15.Theorem 8.
TheSOII and SOIIred indices for PETIM are as follows:(1)
SOIIG=242n+122n+3−2n+3−92n+9130(2)
SOIIredG=122n+122n+3−2n+3−92n+910Proof.
From the edge partitioning based on degree of PETIM, we have following computations forSOII and SOIIred indices:(19)SOIIG=∑ij∈EGdi2+dj2=12+222n+1×22+222n+4−18×22+3262n−6=242n+122n+3−2n+3−92n+9130,(20)SOIIredG=∑ij∈EGdi−12+dj−121−12+2−122n+1×2−12+2−122n+4−18×2−12+3−1262n−6=122n+122n+3−2n+3−92n+910.


### 3.7. Zinc-Porphyrin DendrimerDPZn

Figure10 shows the algebraic graph of the zinc-porphyrin dendrimer DPZn. Based on the degree of end vertices, there are six types of edges in the edge set of DPZn. Table 5 provides the degree-based partitioning of the edges of the DPZn. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 11.Figure 10
Zinc-porphyrin dendrimer (DPZn).Table 5
Edge partitioning based on degree ofDPZn.
EEdi,djFrequencyE1E2,2162n−4E2E2,3402n−16E3E3,382n−16E4E3,44Figure 11
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 9.
TheSO and SOred indices for DPZn are as follows:(1)
SOG=20+562n−12+1652n−113(2)
SOredG=482n−92+1652n−15+413Proof.
From the edge partitioning based on degree ofDPZn, we have following computations for SO and SOred indices:(21)SOG=∑ij∈EGdi2+dj2=22+22162n−4+22+32402n−16+32+3282n−16+32+424=20+562n−12+1652n−113,(22)SOredG=∑ij∈EGdi−12+dj−12=2−12+2−12162n−4+2−12+3−12402n−16+3−12+3−1282n−16+3−12+4−124=482n−92+1652n−15+413.Theorem 10.
TheSOII and SOIIred indices for DPZn are as follows:(1)
SOIIG=1792022n−22n+126(2)
SOIIredG=2564022n−532n+9130Proof.
From the edge partitioning based on degree ofDPZn, we have following computations for SOII and SOIIred indices:(23)SOIIG=∑ij∈EGdi2+dj2=22+22162n−4×22+32402n−16×32+3282n−16×32+424=1792022n−22n+126,(24)SOIIredG=∑ij∈EGdi−12+dj−12=2−12+2−12162n−4×2−12+3−12402n−16×3−12+3−1282n−16×3−12+4−124=2564022n−532n+9130.


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

The algebraic graph of PETAA is shown in Figure12. For PETAA, VPETAA=442n−18 and EPETAA=442n−19. There are six type of edges in the edge set of PETAA, based on the degree of end vertices. Degree-based partition of edges of PETAA is given in Table 6. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 13.Figure 12
Poly (EThyleneAmidoAmine) (PETAA) dendrimer.Table 6
Edge partitioning based on degree of PETAA.
EEdi,djFrequencyE1E1,24.2nE2E1,342n−2E3E2,2162n−8E4E2,3202n−9Figure 13
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 11.
TheSO and SOred indices for PETAA are as follows:(1)
SOG=162n+1−12+42n5+22n+1−110+202n−913(2)
SOredG=42n+42n+1−1+82n+1−12+202n−95Proof.
From the edge partitioning based on degree of PETAA, we have following computations forSO and SOred indices:(25)SOG=∑ij∈EGdi2+dj2=12+2242n+12+3242n−2+22+22162n−8+22+32202n−9=162n+1−12+42n5+22n+1−110+202n−913,(26)SOredG=∑ij∈EGdi−12+dj−12=1−12+2−1242n+1−12+3−1242n−2+2−12+2−12162n−8+2−12+3−12202n−9=4.2n+42n+1−1+82n+1−12+202n−95.Theorem 12.
TheSOII and SOIIred indices for PETAA are as follows:(1)
SOIIG=12802n2023n+2−4022n+1+202n−922n+2+182n+1−913(2)
SOIIredG=1282n2023n+2−4022n+1+202n−922n+2+182n+1−910Proof.
From the edge partitioning based on degree of PETAA, we have following computations forSOII and SOIIred indices:(27)SOIIG=∑ij∈EGdi2+dj2=12+224.2n×12+3242n−2×22+22162n−8×22+32202n−9=1280.2n2023n+2−4022n+1+202n−922n+2+182n+1−913,(28)SOIIredG=∑ij∈EGdi−12+dj−12=1−12+2−124.2n×1−12+3−1242n−2×2−12+2−12162n−8×2−12+3−12202n−9=1282n2023n+2−4022n+1+202n−922n+2+182n+1−910.


## 3.1. Bismuth Triiodide

m−BiI3 is an inorganic chemical formed by the interaction of iodine and bismuth. It is this compound that sparked interest in subjective inorganic research [20]. m−BiI3 is an excellent inorganic chemical that comes in qualitative inorganic analysis [31]. Bi-adoped glass optical strands have been shown to be one of the most promising dynamic laser media. Bi-adoped fibre lasers and optical loudspeakers have been built using several types of bi-adoped fibre strands [32]. Layered m−BiI3 gemstones have a three-layered stack structure, with a bismuth atom plane sandwiched between iodide particle planes to form a continuous plane. Diamond-shaped m−BiI3 crystals with R-3 symmetry are formed by the periodic superposition of three layers. A hexagonal structure with symmetry is formed by a progressive stack of I-Bi-I layers [8, 33].


## 3.2. Bismuth Triiodide Chainm−BiI3

The molecular graph of the unit cell ofm−BiI3 is shown in Figure 1. From Figure 2, 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 1. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 3.Figure 1
Unit cell (bismuth triiodide).Figure 2
The chain form = 3 (bismuth triiodide).Table 1
Edge partitioning based on degree ofm−BiI3m,n.
EEdi,djFrequencyE1E1,64m+8E2E2,620m+4Figure 3
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 1.
TheSO and SOred indices for m−BiI3 are as follows:(1)
SOG=4m+837+220m+410(2)
SOredG=40+20m+45m+126Proof.
From the edge partitioning based on degree ofm−BiI3, we have following computations for SO and SOred indices:(5)SOG=∑ij∈EGdi2+dj2=12+624m+8+22+6220m+4=4m+837+220m+410,(6)SOredG=∑ij∈EGdi−12+dj−12=1−12+6−124m+8+2−12+6−1220m+4=40+20m+45m+126.Theorem 2.
TheSOII and SOIIred indices for m−BiI3 are as follows:(1)
SOIIG=280m2+176m+32370(2)
SOIIredG=320m5m+126Proof.
From the edge partitioning based on degree ofm−BiI3, we have following computations for SOII and SOIIred indices:(7)SOIIG=∏ij∈EGdi2+dj2=12+624m+8×22+6220m+4=280m2+176m+32370,(8)SOIIredG=∏ij∈EGdi−12+dj−12=1−12+6−124m+8×2−12+6−1220m+4=320m5m+126.


## 3.3. Bismuth Triiodide SheetBiI3m×n

The molecular graph of the bismuth triiodide sheetBiI3m×n is shown in Figure 4. It can be observed from Figure 3 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 given in Table 2. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 5.Figure 4
Bismuth triiodide sheet.Table 2
Edge partitioning based on degree ofm−BiI3m×n.
EEdi,djFrequencyE1E1,64m+n+1E2E2,643mn+2m+2n−1E3E3,66nm−1Figure 5
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 3.
TheSO and SOred indices for m−BiI3m×n are as follows:(1)
SOG=4m+n+137+83mn+2m+2n−110+18nm−15(2)
SOredG=20m+n+1+43mn+2m+2n−126+6nm−129Proof.
From the edge partitioning based on degree ofm−BiI3m×n, we have following computations for SO and SOred indices:(9)SOG=∑ij∈EGdi2+dj2=412+62m+n+1+422+623mn+2m+2n−1+32+626nm−1=4m+n+137+83mn+2m+2n−110+18nm−15,(10)SOredG=∑ij∈EGdi−12+dj−12=41−12+6−12m+n+1+42−12+6−123mn+2m+2n−1+6nm−13−12+6−12=20m+n+1+43mn+2m+2n−126+6nm−129.Theorem 4.
TheSOII and SOIIred indices for m−BiI3m×n are as follows:(1)
SOIIG=28803m3n+2m3+4m2n−m2+3m2n2−4mn2−6mn−2m−2n2−n+174(2)
SOIIredG=4803m3n+2m3+4m2n−m2+3m2n2−3mn2−6mn−2m−2n2−n+1754Proof.
From the edge partitioning based on degree ofm−BiI3m×n, we have following computations for SOII and SOIIred indices:(11)SOIIG=∑ij∈EGdi2+dj2=412+62m+n+1×422+623mn+2m+2n−1×6nm−132+62)=28803m3n+2m3+4m2n−m2+3m2n2−4mn2−6mn−2m−2n2−n+174,(12)SOIIredG=∑ij∈EGdi−12+dj−12=41−12+6−12m+n+1×42−12+6−123mn+2m+2n−1×6nm−13−12+6−12=4803m3n+2m3+4m2n−m2+3m2n2−3mn2−6mn−2m−2n2−n+1754.


## 3.4. Dendrimers

The structure of medicine is represented mathematically as an undirected graph, with each vertex representing a molecule and each edge representing an atom-atom connection [34]. Every year, a large number of new medications are developed. Furthermore, selecting the pharmacological substance and organic properties of these new medications necessitates a significant amount of effort and subsequent duties become increasingly detailed and grouped. To test the exhibitions and the responses of new medications, enough reagent rigs and accomplices are required [35, 36]. Nonetheless, in impoverished countries and communities (for example, particular metropolitan networks and countries in South America, Southeast Asia, Africa, and India), there is insufficient funding to purchase reagents and equipment that can be used to assess biochemical attributes. Because of their effects against prion disorders, Alzheimer’s disease, inflammation, HIV, herpes simplex virus (HSV), bacteria, and cancer, dendrimers are interested in medication research. We use the term “topological research of dendrimers” to refer to the study of dendrimers on a topological level [29, 30].


## 3.5. Porphyrin DendrimerDnPn

The algebraic graph ofDnPn is shown in Figure 6. For DnPn, VDnPn=96n−10 and EDnPn=105n−11. Based on the degree of end vertices, there are six types of edges in the edge set of DnPn. Degree-based partition of edges of DnPn is given in Table 3. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 7.Figure 6
Porphyrin dendrimerDnPn.Table 3
Edge partitioning based on degree ofDnPn.
EEdi,djFrequencyE1E1,32nE2E1,4124nE3E2,210n−5E4E2,348n−6E5E3,313nE6E3,48nFigure 7
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 5.
TheSO and SOred indices for DnPn are as follows:(1)
SOG=40n+2n10+12417+59n−102+48n−613s(2)
SOredG=376n+36n−52+48n−65+8n13Proof.
From the edge partitioning based on degree ofDnPn, we have following computations for SO and SOred indices:(13)SOG=∑ij∈EGdi2+dj2=12+322n+12+42124n+22+2210n−5+22+3248n−6+32+3213n+32+428n=40n+2n10+12417+59n−102+48n−613,(14)SOredG=∑ij∈EGdi−12+dj−12=1−12+3−122n+1−12+4−12124n+2−12+2−1210n−5+2−12+3−1248n−6+3−12+3−1213n+3−12+4−128n=376n+36n−52+48n−65+8n13.Theorem 6.
TheSOII and SOIIred indices for DnPn are as follows:(1)
SOIIG=408n2832n2−834n+601105(2)
SOIIredG=1408n1728n2−456n+30165Proof.
From the edge partitioning based on degree ofDnPn, we have following computations for SOII and SOIIred indices:(15)SOIIG=∑ij∈EGdi2+dj2=12+322n×12+42124n×22+2210n−5×22+3248n−6×32+3213n×32+428n=408n2832n2−834n+601105,(16)SOIIredG=∑ij∈EGdi−12+dj−12=1−12+3−122n×1−12+4−12124n×2−12+2−1210n−5×2−12+3−1248n−6×3−12+3−1213n×3−12+4−128n=1408n1728n2−456n+30165.


## 3.6. Propyl Ether Imine (PETIM) Dendrimer

The algebraic graph of PETIM is shown in Figure8. For PETIM, VPETIM=242n−23 and EPETIM=242n−24. There are six type of edges in the edge set of PETIM, based on the degree of end vertices. Degree-based partition of edges of PETIM is given in Table 4. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 9.Figure 8
Propyl ether imine (PETIM) dendrimer.Table 4
Edge partitioning based on degree of PETIM.
EEdi,djFrequencyE1E1,22n+1E2E2,22n+4−18E3E2,362n−6Figure 9
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 7.
TheSO and SOred indices for PETIM are as follows:(1)
SOG=2n+15+42n+3−92+62n−113(2)
SOredG=2n+1+22n+3−92+62n−15Proof.
From the edge partitioning based on degree of PETIM, we have following computations forSO and SOred indices:(17)SOG=∑ij∈EGdi2+dj2=12+222n+1+22+222n+4−18+22+3262n−6=2n+15+42n+3−92+62n−113,(18)SOredG=∑ij∈EGdi−12+dj−12=1−12+2−122n+1+2−12+2−122n+4−18+2−12+3−1262n−6=2n+1+22n+3−92+62n−15.Theorem 8.
TheSOII and SOIIred indices for PETIM are as follows:(1)
SOIIG=242n+122n+3−2n+3−92n+9130(2)
SOIIredG=122n+122n+3−2n+3−92n+910Proof.
From the edge partitioning based on degree of PETIM, we have following computations forSOII and SOIIred indices:(19)SOIIG=∑ij∈EGdi2+dj2=12+222n+1×22+222n+4−18×22+3262n−6=242n+122n+3−2n+3−92n+9130,(20)SOIIredG=∑ij∈EGdi−12+dj−121−12+2−122n+1×2−12+2−122n+4−18×2−12+3−1262n−6=122n+122n+3−2n+3−92n+910.


## 3.7. Zinc-Porphyrin DendrimerDPZn

Figure10 shows the algebraic graph of the zinc-porphyrin dendrimer DPZn. Based on the degree of end vertices, there are six types of edges in the edge set of DPZn. Table 5 provides the degree-based partitioning of the edges of the DPZn. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 11.Figure 10
Zinc-porphyrin dendrimer (DPZn).Table 5
Edge partitioning based on degree ofDPZn.
EEdi,djFrequencyE1E2,2162n−4E2E2,3402n−16E3E3,382n−16E4E3,44Figure 11
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 9.
TheSO and SOred indices for DPZn are as follows:(1)
SOG=20+562n−12+1652n−113(2)
SOredG=482n−92+1652n−15+413Proof.
From the edge partitioning based on degree ofDPZn, we have following computations for SO and SOred indices:(21)SOG=∑ij∈EGdi2+dj2=22+22162n−4+22+32402n−16+32+3282n−16+32+424=20+562n−12+1652n−113,(22)SOredG=∑ij∈EGdi−12+dj−12=2−12+2−12162n−4+2−12+3−12402n−16+3−12+3−1282n−16+3−12+4−124=482n−92+1652n−15+413.Theorem 10.
TheSOII and SOIIred indices for DPZn are as follows:(1)
SOIIG=1792022n−22n+126(2)
SOIIredG=2564022n−532n+9130Proof.
From the edge partitioning based on degree ofDPZn, we have following computations for SOII and SOIIred indices:(23)SOIIG=∑ij∈EGdi2+dj2=22+22162n−4×22+32402n−16×32+3282n−16×32+424=1792022n−22n+126,(24)SOIIredG=∑ij∈EGdi−12+dj−12=2−12+2−12162n−4×2−12+3−12402n−16×3−12+3−1282n−16×3−12+4−124=2564022n−532n+9130.


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

The algebraic graph of PETAA is shown in Figure12. For PETAA, VPETAA=442n−18 and EPETAA=442n−19. There are six type of edges in the edge set of PETAA, based on the degree of end vertices. Degree-based partition of edges of PETAA is given in Table 6. Also, the graphical comparison of SO,SOred,SOII, and SOIIred indices is shown in Figure 13.Figure 12
Poly (EThyleneAmidoAmine) (PETAA) dendrimer.Table 6
Edge partitioning based on degree of PETAA.
EEdi,djFrequencyE1E1,24.2nE2E1,342n−2E3E2,2162n−8E4E2,3202n−9Figure 13
Graphical comparison ofSO,SOred,SOII, and SOIIred indices.Theorem 11.
TheSO and SOred indices for PETAA are as follows:(1)
SOG=162n+1−12+42n5+22n+1−110+202n−913(2)
SOredG=42n+42n+1−1+82n+1−12+202n−95Proof.
From the edge partitioning based on degree of PETAA, we have following computations forSO and SOred indices:(25)SOG=∑ij∈EGdi2+dj2=12+2242n+12+3242n−2+22+22162n−8+22+32202n−9=162n+1−12+42n5+22n+1−110+202n−913,(26)SOredG=∑ij∈EGdi−12+dj−12=1−12+2−1242n+1−12+3−1242n−2+2−12+2−12162n−8+2−12+3−12202n−9=4.2n+42n+1−1+82n+1−12+202n−95.Theorem 12.
TheSOII and SOIIred indices for PETAA are as follows:(1)
SOIIG=12802n2023n+2−4022n+1+202n−922n+2+182n+1−913(2)
SOIIredG=1282n2023n+2−4022n+1+202n−922n+2+182n+1−910Proof.
From the edge partitioning based on degree of PETAA, we have following computations forSOII and SOIIred indices:(27)SOIIG=∑ij∈EGdi2+dj2=12+224.2n×12+3242n−2×22+22162n−8×22+32202n−9=1280.2n2023n+2−4022n+1+202n−922n+2+182n+1−913,(28)SOIIredG=∑ij∈EGdi−12+dj−12=1−12+2−124.2n×1−12+3−1242n−2×2−12+2−12162n−8×2−12+3−12202n−9=1282n2023n+2−4022n+1+202n−922n+2+182n+1−910.


## 4. Applications

Bismuth compounds have been utilised in medicine, especially for the treatment of gastrointestinal problems. Bismuth contains antimicrobial, antileishmanial, and anticancer characteristics in addition to its well-known gastroprotective benefits and efficacy in treatingH. pylori infection. Bismuth compound have also been used for the treatment of war wounds, cholera, and gastroenteritis. Bi-adoped glass optical strands have been shown to be one of the most promising dynamic laser media. Bi-adoped fibre lasers and optical loudspeakers have been built using several types of Bi-adoped fibre strands. Dendrimers are interesting for biomedical applications because of their effects against prion disorders, Alzheimer’s disease, inflammation, HIV, herpes simplex virus (HSV), bacteria, and cancer. In our work, aspects of the biological chemistry of bismuth and dendrimers are discussed, and biomolecular targets associated with their treatment are highlighted. This review strives to provide the reader with an up to date account of bismuth-based drugs as well as dendrimers currently used to treat patients and potential medicinal applications of these drugs.


## 5. Conclusion

In medical science, chemical, medical, biological, and pharmaceutical properties of molecular structure are essential for drug design. Various tools of mathematics are used to predict different properties of chemical compounds like topological index. We can associate a single number with a molecular graph of a chemical complex using the topological index. Drugs and other chemical substances are frequently shown as polygonal shapes, trees, graphs, and other geometrical shapes. In this study, we discuss the newly introduced Sombor invariants for bismuth (III) iodide and dendrimers. Also, we presented the graphical comparison for the different versions of the computed TIs. The study of chemical characteristics and bonding of Sombor indices is an interesting subject for researchers [37].



---

*Source: 3163294-2022-07-08.xml*