# Analysis of Dendrimer Generation by Sombor Indices

Authors: Shahid Amin; Abaid Ur Rehman Virk; M.A. Rehman; Nehad Ali Shah
Journal: Journal of Chemistry
 (2021)

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## Abstract

Dendrimers are highly branched, star-shaped macromolecules with nanometer-scale dimensions. Dendrimers are defined by three components: a central core, an interior dendritic structure (the branches), and an exterior surface with functional surface groups. Topological indices are numerical numbers that help us to understand the topology of different dendrimers and can be used to predict the properties without performing experiments in the wet lab. In the present paper, we computed the Sombor index and the reduced version of the Sombor index for the molecular graphs of phosphorus-containing dendrimers, porphyrin-cored dendrimers, PDI-cored dendrimers, triazine-based dendrimers, and aliphatic polyamide dendrimers. We also plotted our results by using Maple 2015 which help us to see the dependence of the Sombor index and reduced Sombor index on the involved parameters. Our results may help to develop better understanding about phosphorus-containing dendrimers, porphyrin-cored dendrimers, PDI-cored dendrimers, triazine-based dendrimers, and aliphatic polyamide dendrimers. Our results are also useful in the pharmaceutical industry and drug delivery.


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## Body

## 1. Introduction

Dendrimers are highly branched star-molded macromolecules with nanometer-scale measurements [1]. A dendrimer consists of three modules: a central core, an interior surface (branches), and the outer surface. A functional surface group is attached with the outer core. Various blends of these parts yield results of various shapes and sizes with protected inside centers that are an ideal contender for applications in both organic and materials sciences [2]. The characteristics of a dendrimer depend on the external group attached with the outer surface. Dendrimers have acquired a wide scope of uses in supramolecular science, especially in drug delivery, gene transfection, catalysis, energy harvesting, photo activity, molecular weight and size determination, rheology modification, and nanoscale science and technology. A dendrimer acts as a solubilizing agent in different reactions. Dendrimers have a wide range of applications in different fields of sciences [3]. The construction of dendrimers is presented in Figure 1.Figure 1
Modules of the dendrimer.Mathematical chemistry is the branch of mathematics in which mathematical tools are used to solve the problems arising in chemistry [4]. One of these tools is graphical representation of chemical compounds, and this representation is known as the molecular graph of the concerned chemical compound [5]. In the molecular graph of a chemical compound, atoms are represented as vertices, and bounds are represented as edges [6]. Topological invariants of molecular graphs are numerical numbers that enable us to collect information about concerned chemical structure and give us its hidden properties without performing experiments [7–11]. The first topological index was put forward by Wiener in 1947 [12] when he was trying to find the boiling points of alkane. This discovery led to the beginning of the theory of topological indices. The first degree-based topological index was put forward by Randić in 1975 [13]. After the success of the Randić index, Gutman introduced the Zagreb indices. There are hundreds of topological indices present in the literature [14–18]. Recently, Gutman, in 2021 [19], defined the idea of Sombor indices. Sombor indices have two variants, Sombor index and reduced Sombor index, as follows:(1)SOG=∑ij∈EGdi2+dj2,SOredG=∑ij∈EGdi−12+dj−12.The aim of this paper is to study the phosphorus-containing dendrimers, porphyrin-cored dendrimers, PDI-cored dendrimers, triazine-based dendrimers, and aliphatic polyamide dendrimers. We computed the Sombor index and the reduced version of the Sombor index for the aforementioned dendrimers. We also present graphical representations of our results to see the dependence of computed indices on the involved parameters.Throughout this paper, we consider all graphs to be simple and connected, anddu denotes the degree of vertex u which is equal to the number of vertices at distance one to it. For the notation used in this paper but not defined, we refer to in [20, 21] and references therein.


## 2. Methodology

Firstly, we obtain the molecular graphs of phosphorus-containing dendrimers, porphyrin-cored dendrimers, PDI-cored dendrimers, triazine-based dendrimers, and aliphatic polyamide dendrimers. Secondly, we compute the order and size of these molecular graphs and classify their edge sets and vertex sets into different classes with respect to the degrees of vertices. Thirdly, we compute the Sombor and reduced Sombor indices for the molecular graphs of phosphorus-containing dendrimers, porphyrin-cored dendrimers, PDI-cored dendrimers, triazine-based dendrimers, and aliphatic polyamide dendrimers. Lastly, we plot our obtained results by using Maple 2015 software.


## 3. Main Results

In this section, we present Sombor and reduced Sombor indices for phosphorus-containing dendrimers, porphyrin-cored dendrimers, PDI-cored dendrimers, triazine-based dendrimers, and aliphatic polyamide dendrimers.


### 3.1. Phosphorus-Containing Dendrimers

Phosphorus-containing dendrimers have functionalities with pendant nitroxyl radicals, and these radicals show a solid attractive trade interaction. Let us considerD1m to be the molecular graph of phosphorus-containing dendrimers, where m shows the generation stage of D1m. Figure 2 shows the molecular graph D1m of phosphorus-containing dendrimers.Figure 2
D1m for m = 3.From Figure2, we can observe that the order and size of D1m are 911×2m+1−8 and 69×2m+2−13, respectively. If VD1m is the vertex set, then from Figure 2, we can classify VD1m into four subsets V1, V2, V3, and V4 of vertices of degrees 1, 2, 3, and 4 such that VD1m=V1D1m+V2D1m+V3D1m+V4D1m. The cardinalities of V1D1m, V2D1m, V3D1m, and V4D1m are 42×2m−12, 96×2m−39, 42×2m−18, and 18×2n−3, respectively.IfED1m represents the edge set, then Figure 2 shows that there are the following six different types of edges with respect to the degrees of end vertices present in the molecular graph of D1m:(2)ε1D1m=ε1,3D1m=e=uv∈εD1m:du=1,dv=3,ε2D1m=ε1,4D1m=e=uv∈εD1m:du=1,dv=4,ε3D1m=ε2,2D1m=e=uv∈εD1m:du=2,dv=2,ε4D1m=ε2,3D1m=e=uv∈εD1m:du=2,dv=3,ε5D1m=ε2,4D1m=e=uv∈εD1m:du=2,dv=4,ε6D1m=ε3,4D1m=e=uv∈εD1m:du=3,dv=4.Table1 explains the edge partition of the edge set of D1m in detail.Table 1
Degree-based edge partition ofD1m.
εεdu,dvFrequencyε1ε1,362m−1ε2ε1,465×2m−1ε3ε2,2182m+1−1ε4ε2,362m+4−7ε5ε2,43×2m+3ε6ε3,463×2m−1Frequency means the total number of edges in the particular class.Theorem 1.
TheSO and SOred indices for D1m are as follows:(i)
SOG=21/2+m5−625+72×21/2+m+4852m+96132m+30172m−362−4213−617+90×2m−30(ii)
SOredG=24×21/2+m5+18132m+3621/2+m+9652m+282×2m−182−613−425−66Proof.
For the edge partition of the vertex set ofD1m, we have the following computations for SO and SOred indices:(3)SOD1=∑ijdi2+dj2=12+3262m−1+12+4265×2m−1+22+22182m+1−1+22+3262m+4−7+22+423×2m+3+32+4263×2m−1=21/2+m5−610+72×21/2+m+4852m+96132m+30172m−362−4213−617+902m−30,SOredD1=∑ijdi−12+dj−12=1−12+3−1262m−1+1−12+4−1265×2m−1+2−12+2−12182m+1−1+2−12+3−1262m+4−7+2−12+4−123×2m+3+3−12+4−1263×2m−124×21/2+m5+18132m+36×21/2+m+9652m+282×2m−182−613−425−66.


### 3.2. Porphyrin-Cored Dendrimers

Figure3 shows the molecular graph D2m of porphyrin-cored dendrimers, where m represents different generations of D2m.Figure 3
D2m for m = 1.From Figure3, we can observe that the order and size of D2m are 42m+3+9 and 42m+3+11, respectively. If VD2m is the vertex set, then from Figure 3, we can classify this vertex set into four subsets V1D2m, V2D2m, V3D2m, and V4D2m with respect to degrees such that VD2m=V1D2m+V2D2m+V3D2m+V4D2m. The cardinalities of V1D2m, V2D2m, V3D2m, and V4D2m are 12×2m−8, 12×2m+32, 4×2m+16, and 4×2n−4 vertices, respectively.IfED2m represents the edge set, then Figure 3 shows that there are the following eight different types of edges present in the molecular graph of D2m:(4)E1D2m=E1,2D2m=e=uv∈ED2m:du=1,dv=2,E2D2m=E1,3D2m=e=uv∈ED2m:du=1,dv=3,E3D2m=E1,4D2m=e=uv∈ED2m:du=1,dv=4,E4D2m=E2,2D2m=e=uv∈ED2m:du=2,dv=2,E5D2m=E2,3D2m=e=uv∈ED2m:du=2,dv=3,E6D2m=E2,4D2m=e=uv∈ED2m:du=2,dv=4,E7D2m=E3,3D2m=e=uv∈ED2m:du=3,dv=3,E8D2m=E3,4D2m=e=uv∈ED2m:du=3,dv=4.Table2 gives the detailed explanation about the edge partition of the edge set of D1m.Table 2
Degree-based edge partition ofD2m.
EEdu,dvFrequencyE1E1,24×2mE2E1,34×2m−4E3E1,44×2m−4E4E2,24×2m+20E5E2,34×2m+32E6E2,48×2m−8E7E3,312E8E3,44×2m−4Theorem 2.
TheSO and SOred indices for D2m are as follows:(i)
SOG=4×21/2+m5+2052m−425+821/2+m+4172m+4132m−165+202m+762−417+3213−20(ii)
SOredG=8×21/2+m5+452m−825+4×21/2+m+4132m+325+242m+442−413−20Proof.
From the edge partition ofD2m given in Table 2, we have the following computations of SO and SOred indices:(5)SOD2=∑ijdi2+dj2=12+224×2m+12+324×2m−4+12+424×2m−4+22+224×2m+20+22+324×2m+32+22+428×2m−8+32+3212+32+424×2m−4=4×21/2+m5+2052m−425+821/2+m+4172m+4132m−165+202m+762−417+3213−20,SOredD2=∑ijdi−12+dj−12=1−12+2−124×2m+1−12+3−124×2m−4+1−12+424×2m−4+2−12+2−124×2m+20+2−12+3−124×2m+32+2−12+4−128×2m−8+3−12+3−1212+3−12+4−124×2m−4=8×21/2+m5+452m−825+4×21/2+m+4132m325+242m+442−413−20.


### 3.3. PDI-Cored Dendrimers

The water-dissolvable PDI-cored dendrimers have various accommodations, containing low cytotoxicity, solid red fluorescence, high quantum yield, amazing photostability, and flexible surface alteration. These dendrimers have numerous applications in different fields such as fluorescence live-cell imaging and labeling. LetD3m be the molecular graph of PDI-cored dendrimers; then, Figure 4 shows the 2D graph of D3m.Figure 4
D3m for m = 1.From Figure4, we can observe that the order and size of D3m are 20×2m+20 and 20×22m+20, respectively. If VD3m is the vertex set, then by observing Figure 4, we can classify this vertex set into three subsets V1D3m, V2D3m, and V3D3m such that VD3m=V1D3m+V2D3m+V3D3m. The cardinalities of V1D3m, V2D3m, and V3D3m are 2×2m+1+4, 6×2m+1, and 2×2m+1+16, respectively.IfED3m represents the edge set, then Figure 4 shows that there are the following five different types of edges present in the molecular graph of D3m:(6)E1D3m=E1,2D3m=e=uv∈ED3m:du=1,dv=2,E2D3m=E1,3D3m=e=uv∈ED3m:du=1,dv=3,E3D3m=E2,2D3m=e=uv∈ED3m:du=2,dv=2,E4D3m=E2,3D3m=e=uv∈ED3m:du=2,dv=3,E5D3m=E3,3D3m=e=uv∈ED3m:du=3,dv=3.Table3 gives the detailed explanation about the edge partition of the edge set of D3m.Table 3
Degree-based edge partition ofD3m.
EEdu,dvFrequencyE1E1,22m+1E2E1,342m−1+1E3E2,23×2m+1+1E4E2,320×2m−1E5E3,322Theorem 3.
TheSO and SOred indices for D3m are as follows:(i)
SOD3=23/2+m5+5×2m+1+12×21/2+m+10132m+410+682(ii)
SOredD3=6×21/2+m+1052m+6×2m+452+8Proof.
From the edge partition ofD3m given in Table 3, we have the following computations of SO and SOred indices:(7)SOD3=∑ijdi2+dj2+22+223×2m+1+1=12+222m+1+12+3242m−1+1+22+3220×2m−1+32+3222=23/2+m5+5×2m+1+12×21/2+m+10132m+410+682,SOredD3=∑ijdi−12+dj−12=1−12+2−122m+1+1−12+3−1242m−1+1+2−12+2−123×2m+1+1+2−12+3−1220×2m−1+3−12+3−1222=6×21/2+m+1052m+6×2m+452+8.


### 3.4. Triazine-Based Dendrimers

The divergent method is used for the synthesis of triazine-based dendrimers. Triazine-based dendrimers are less toxic and can be further studied as drug carriers. LetD4m represent the molecular graph of triazine-based dendrimer drug carriers in the future. Figure 5 shows the molecular graph of D4m.Figure 5
D4m for m = 2.From Figure5, we can observe that the order and size of D3m are 25×22m+2+1/3 and 7×22m+1+1, respectively. If VD4m is the vertex set, then by observing Figure 5, we can classify this vertex set into four subsets V1D4m, V2D4m, and V3D4m such that VD4m=V1D4m+V2D4m+V3D4m. The cardinalities of V1D4m, V2D4m, and V3D4m are 22m+1, 22m+1+7×4m+1/6+4m+1/3, and 4+5×4m+1/6−10/3, respectively.IfED4m represents the edge set, then Figure 5 shows that there are the following four different types of edges present in the molecular graph of D4m:(8)E1D4m=E1,2D4m=e=uv∈ED4m:du=1,dv=2,E2D4m=E2,2D4m=e=uv∈ED4m:du=2,dv=2,E3D4m=E2,3D4m=e=uv∈ED4m:du=2,dv=3,E4D4m=E3,3D4m=e=uv∈ED4m:du=3,dv=3.Table4 gives the detailed explanation about the edge set of D4m.Table 4
Degree-based edge partition ofD4m.
EEdu,dvFrequencyE1E1,22m+1E2E2,225×22m−2/3E3E2,3211×22m+4/3E4E3,32×22m+1−1/3Theorem 4.
TheSO and SOred indices for D4m are as follows:(i)
SOD4=52m+1+32/324m−14/32+22/3132m+8/313(ii)
SOredD4=4×2m+624m−8/32+22/352m+8/35Proof.
From the edge partition of the edge set ofD4m given in Table 4, we have the following computations for SO and SOred indices:(9)SOD4∑ijdi2+dj2+22+223×2m+1+1=12+222m+1+22+2225×22m−23+22+32211×22m+43+32+322×22m+1−1352m+1+32324m−1432+223132m+8313,SOredD4=∑ijdi−12+dj−12=1−12+2−122m+1+2−12+2−1225×22m−23+2−12+3−12211×22m+43+3−12+3−122×22m+1−134×2m+624m−832+22352m+835.


### 3.5. Aliphatic Polyamide Dendrimers

Recently, Jishkariani, for the first time, studied aliphatic polyamide dendrimers containing ethylenediamine and piperazine. These dendrimers are enzymatically and hydrolytically stable. LetD5m represent the molecular graph of the aliphatic polyamide-based dendrimer. Figure 6 shows the molecular graph of D5m.Figure 6
D5m for m = 4.From Figure6, we can observe that the order and size of D5m are 22m+3−5. If VD5m is the vertex set, then by observing Figure 6, we can classify this vertex set into three subsets V1D5m, V2D5m, and V3D5m such that VD5m=V1D5m+V2D5m+V3D5m+V4D5m. The cardinalities of V1D5m, V2D5m, and V3D5m are 43×2m−1−1, 43×2m−1−1, and 2m+122m−1, respectively.IfED5m represents the edge set, then Figure 6 shows that there are the following four different types of edges present in the molecular graph of D5m:(10)E1D5m=E1,2D5m=e=uv∈ED5m:du=1,dv=2,E2D5m=E2,3D5m=e=uv∈ED5m:du=2,dv=3,E3D5m=E1,3D5m=e=uv∈ED5m:du=1,dv=3,E4D5m=E1,4D5m=e=uv∈ED5m:du=1,dv=4.Table5 gives the detailed explanation about the edge set of D5m.Table 5
Degree-based edge partition ofD5m.
EEdu,dvFrequencyE1E1,22m+1E2E2,32m+1E3E1,322m−1E4E1,422m−1E5E2,222m−1E6E3,422m−1E7E3,32E8E2,42m+2−4Theorem 5.
TheSO and SOred indices for D5m are as follows:(i)
SOD5=23/2+m5+1052m+2m+117−210+132m+1+4×21/2+m−217−85+22+10×2m−10(ii)
SOredD5=6132m+52m+1+23/2+m+18×2m+22−613−14Proof.
From the edge partition of the edge set ofD5m given in Table 5, we have the following computations of SO and SOred indices:(11)SOD5=∑ijdi2+dj2+22+223×2m+1+1=12+222m+1+22+322m+1+12+3222m−1+12+4222m−1+22+2222m−1+32+4222m−132+322+22+422m+2−4=23/2+m5+1052m+2m+117−210+132m+1+4×21/2+m−217−85+22+10×2m−10,SOredD5=∑ijdi−12+dj−12=1−12+2−122m+1+2−12+3−122m+1+1−12+3−1222m−1+1−12+4−1222m−1+2−12+2−1222m−1+3−12+4−1222m−13−12+3−122+2−12+4−122m+2−4=6132m+52m+1+23/2+m+18×2m+22−613−14.


## 3.1. Phosphorus-Containing Dendrimers

Phosphorus-containing dendrimers have functionalities with pendant nitroxyl radicals, and these radicals show a solid attractive trade interaction. Let us considerD1m to be the molecular graph of phosphorus-containing dendrimers, where m shows the generation stage of D1m. Figure 2 shows the molecular graph D1m of phosphorus-containing dendrimers.Figure 2
D1m for m = 3.From Figure2, we can observe that the order and size of D1m are 911×2m+1−8 and 69×2m+2−13, respectively. If VD1m is the vertex set, then from Figure 2, we can classify VD1m into four subsets V1, V2, V3, and V4 of vertices of degrees 1, 2, 3, and 4 such that VD1m=V1D1m+V2D1m+V3D1m+V4D1m. The cardinalities of V1D1m, V2D1m, V3D1m, and V4D1m are 42×2m−12, 96×2m−39, 42×2m−18, and 18×2n−3, respectively.IfED1m represents the edge set, then Figure 2 shows that there are the following six different types of edges with respect to the degrees of end vertices present in the molecular graph of D1m:(2)ε1D1m=ε1,3D1m=e=uv∈εD1m:du=1,dv=3,ε2D1m=ε1,4D1m=e=uv∈εD1m:du=1,dv=4,ε3D1m=ε2,2D1m=e=uv∈εD1m:du=2,dv=2,ε4D1m=ε2,3D1m=e=uv∈εD1m:du=2,dv=3,ε5D1m=ε2,4D1m=e=uv∈εD1m:du=2,dv=4,ε6D1m=ε3,4D1m=e=uv∈εD1m:du=3,dv=4.Table1 explains the edge partition of the edge set of D1m in detail.Table 1
Degree-based edge partition ofD1m.
εεdu,dvFrequencyε1ε1,362m−1ε2ε1,465×2m−1ε3ε2,2182m+1−1ε4ε2,362m+4−7ε5ε2,43×2m+3ε6ε3,463×2m−1Frequency means the total number of edges in the particular class.Theorem 1.
TheSO and SOred indices for D1m are as follows:(i)
SOG=21/2+m5−625+72×21/2+m+4852m+96132m+30172m−362−4213−617+90×2m−30(ii)
SOredG=24×21/2+m5+18132m+3621/2+m+9652m+282×2m−182−613−425−66Proof.
For the edge partition of the vertex set ofD1m, we have the following computations for SO and SOred indices:(3)SOD1=∑ijdi2+dj2=12+3262m−1+12+4265×2m−1+22+22182m+1−1+22+3262m+4−7+22+423×2m+3+32+4263×2m−1=21/2+m5−610+72×21/2+m+4852m+96132m+30172m−362−4213−617+902m−30,SOredD1=∑ijdi−12+dj−12=1−12+3−1262m−1+1−12+4−1265×2m−1+2−12+2−12182m+1−1+2−12+3−1262m+4−7+2−12+4−123×2m+3+3−12+4−1263×2m−124×21/2+m5+18132m+36×21/2+m+9652m+282×2m−182−613−425−66.


## 3.2. Porphyrin-Cored Dendrimers

Figure3 shows the molecular graph D2m of porphyrin-cored dendrimers, where m represents different generations of D2m.Figure 3
D2m for m = 1.From Figure3, we can observe that the order and size of D2m are 42m+3+9 and 42m+3+11, respectively. If VD2m is the vertex set, then from Figure 3, we can classify this vertex set into four subsets V1D2m, V2D2m, V3D2m, and V4D2m with respect to degrees such that VD2m=V1D2m+V2D2m+V3D2m+V4D2m. The cardinalities of V1D2m, V2D2m, V3D2m, and V4D2m are 12×2m−8, 12×2m+32, 4×2m+16, and 4×2n−4 vertices, respectively.IfED2m represents the edge set, then Figure 3 shows that there are the following eight different types of edges present in the molecular graph of D2m:(4)E1D2m=E1,2D2m=e=uv∈ED2m:du=1,dv=2,E2D2m=E1,3D2m=e=uv∈ED2m:du=1,dv=3,E3D2m=E1,4D2m=e=uv∈ED2m:du=1,dv=4,E4D2m=E2,2D2m=e=uv∈ED2m:du=2,dv=2,E5D2m=E2,3D2m=e=uv∈ED2m:du=2,dv=3,E6D2m=E2,4D2m=e=uv∈ED2m:du=2,dv=4,E7D2m=E3,3D2m=e=uv∈ED2m:du=3,dv=3,E8D2m=E3,4D2m=e=uv∈ED2m:du=3,dv=4.Table2 gives the detailed explanation about the edge partition of the edge set of D1m.Table 2
Degree-based edge partition ofD2m.
EEdu,dvFrequencyE1E1,24×2mE2E1,34×2m−4E3E1,44×2m−4E4E2,24×2m+20E5E2,34×2m+32E6E2,48×2m−8E7E3,312E8E3,44×2m−4Theorem 2.
TheSO and SOred indices for D2m are as follows:(i)
SOG=4×21/2+m5+2052m−425+821/2+m+4172m+4132m−165+202m+762−417+3213−20(ii)
SOredG=8×21/2+m5+452m−825+4×21/2+m+4132m+325+242m+442−413−20Proof.
From the edge partition ofD2m given in Table 2, we have the following computations of SO and SOred indices:(5)SOD2=∑ijdi2+dj2=12+224×2m+12+324×2m−4+12+424×2m−4+22+224×2m+20+22+324×2m+32+22+428×2m−8+32+3212+32+424×2m−4=4×21/2+m5+2052m−425+821/2+m+4172m+4132m−165+202m+762−417+3213−20,SOredD2=∑ijdi−12+dj−12=1−12+2−124×2m+1−12+3−124×2m−4+1−12+424×2m−4+2−12+2−124×2m+20+2−12+3−124×2m+32+2−12+4−128×2m−8+3−12+3−1212+3−12+4−124×2m−4=8×21/2+m5+452m−825+4×21/2+m+4132m325+242m+442−413−20.


## 3.3. PDI-Cored Dendrimers

The water-dissolvable PDI-cored dendrimers have various accommodations, containing low cytotoxicity, solid red fluorescence, high quantum yield, amazing photostability, and flexible surface alteration. These dendrimers have numerous applications in different fields such as fluorescence live-cell imaging and labeling. LetD3m be the molecular graph of PDI-cored dendrimers; then, Figure 4 shows the 2D graph of D3m.Figure 4
D3m for m = 1.From Figure4, we can observe that the order and size of D3m are 20×2m+20 and 20×22m+20, respectively. If VD3m is the vertex set, then by observing Figure 4, we can classify this vertex set into three subsets V1D3m, V2D3m, and V3D3m such that VD3m=V1D3m+V2D3m+V3D3m. The cardinalities of V1D3m, V2D3m, and V3D3m are 2×2m+1+4, 6×2m+1, and 2×2m+1+16, respectively.IfED3m represents the edge set, then Figure 4 shows that there are the following five different types of edges present in the molecular graph of D3m:(6)E1D3m=E1,2D3m=e=uv∈ED3m:du=1,dv=2,E2D3m=E1,3D3m=e=uv∈ED3m:du=1,dv=3,E3D3m=E2,2D3m=e=uv∈ED3m:du=2,dv=2,E4D3m=E2,3D3m=e=uv∈ED3m:du=2,dv=3,E5D3m=E3,3D3m=e=uv∈ED3m:du=3,dv=3.Table3 gives the detailed explanation about the edge partition of the edge set of D3m.Table 3
Degree-based edge partition ofD3m.
EEdu,dvFrequencyE1E1,22m+1E2E1,342m−1+1E3E2,23×2m+1+1E4E2,320×2m−1E5E3,322Theorem 3.
TheSO and SOred indices for D3m are as follows:(i)
SOD3=23/2+m5+5×2m+1+12×21/2+m+10132m+410+682(ii)
SOredD3=6×21/2+m+1052m+6×2m+452+8Proof.
From the edge partition ofD3m given in Table 3, we have the following computations of SO and SOred indices:(7)SOD3=∑ijdi2+dj2+22+223×2m+1+1=12+222m+1+12+3242m−1+1+22+3220×2m−1+32+3222=23/2+m5+5×2m+1+12×21/2+m+10132m+410+682,SOredD3=∑ijdi−12+dj−12=1−12+2−122m+1+1−12+3−1242m−1+1+2−12+2−123×2m+1+1+2−12+3−1220×2m−1+3−12+3−1222=6×21/2+m+1052m+6×2m+452+8.


## 3.4. Triazine-Based Dendrimers

The divergent method is used for the synthesis of triazine-based dendrimers. Triazine-based dendrimers are less toxic and can be further studied as drug carriers. LetD4m represent the molecular graph of triazine-based dendrimer drug carriers in the future. Figure 5 shows the molecular graph of D4m.Figure 5
D4m for m = 2.From Figure5, we can observe that the order and size of D3m are 25×22m+2+1/3 and 7×22m+1+1, respectively. If VD4m is the vertex set, then by observing Figure 5, we can classify this vertex set into four subsets V1D4m, V2D4m, and V3D4m such that VD4m=V1D4m+V2D4m+V3D4m. The cardinalities of V1D4m, V2D4m, and V3D4m are 22m+1, 22m+1+7×4m+1/6+4m+1/3, and 4+5×4m+1/6−10/3, respectively.IfED4m represents the edge set, then Figure 5 shows that there are the following four different types of edges present in the molecular graph of D4m:(8)E1D4m=E1,2D4m=e=uv∈ED4m:du=1,dv=2,E2D4m=E2,2D4m=e=uv∈ED4m:du=2,dv=2,E3D4m=E2,3D4m=e=uv∈ED4m:du=2,dv=3,E4D4m=E3,3D4m=e=uv∈ED4m:du=3,dv=3.Table4 gives the detailed explanation about the edge set of D4m.Table 4
Degree-based edge partition ofD4m.
EEdu,dvFrequencyE1E1,22m+1E2E2,225×22m−2/3E3E2,3211×22m+4/3E4E3,32×22m+1−1/3Theorem 4.
TheSO and SOred indices for D4m are as follows:(i)
SOD4=52m+1+32/324m−14/32+22/3132m+8/313(ii)
SOredD4=4×2m+624m−8/32+22/352m+8/35Proof.
From the edge partition of the edge set ofD4m given in Table 4, we have the following computations for SO and SOred indices:(9)SOD4∑ijdi2+dj2+22+223×2m+1+1=12+222m+1+22+2225×22m−23+22+32211×22m+43+32+322×22m+1−1352m+1+32324m−1432+223132m+8313,SOredD4=∑ijdi−12+dj−12=1−12+2−122m+1+2−12+2−1225×22m−23+2−12+3−12211×22m+43+3−12+3−122×22m+1−134×2m+624m−832+22352m+835.


## 3.5. Aliphatic Polyamide Dendrimers

Recently, Jishkariani, for the first time, studied aliphatic polyamide dendrimers containing ethylenediamine and piperazine. These dendrimers are enzymatically and hydrolytically stable. LetD5m represent the molecular graph of the aliphatic polyamide-based dendrimer. Figure 6 shows the molecular graph of D5m.Figure 6
D5m for m = 4.From Figure6, we can observe that the order and size of D5m are 22m+3−5. If VD5m is the vertex set, then by observing Figure 6, we can classify this vertex set into three subsets V1D5m, V2D5m, and V3D5m such that VD5m=V1D5m+V2D5m+V3D5m+V4D5m. The cardinalities of V1D5m, V2D5m, and V3D5m are 43×2m−1−1, 43×2m−1−1, and 2m+122m−1, respectively.IfED5m represents the edge set, then Figure 6 shows that there are the following four different types of edges present in the molecular graph of D5m:(10)E1D5m=E1,2D5m=e=uv∈ED5m:du=1,dv=2,E2D5m=E2,3D5m=e=uv∈ED5m:du=2,dv=3,E3D5m=E1,3D5m=e=uv∈ED5m:du=1,dv=3,E4D5m=E1,4D5m=e=uv∈ED5m:du=1,dv=4.Table5 gives the detailed explanation about the edge set of D5m.Table 5
Degree-based edge partition ofD5m.
EEdu,dvFrequencyE1E1,22m+1E2E2,32m+1E3E1,322m−1E4E1,422m−1E5E2,222m−1E6E3,422m−1E7E3,32E8E2,42m+2−4Theorem 5.
TheSO and SOred indices for D5m are as follows:(i)
SOD5=23/2+m5+1052m+2m+117−210+132m+1+4×21/2+m−217−85+22+10×2m−10(ii)
SOredD5=6132m+52m+1+23/2+m+18×2m+22−613−14Proof.
From the edge partition of the edge set ofD5m given in Table 5, we have the following computations of SO and SOred indices:(11)SOD5=∑ijdi2+dj2+22+223×2m+1+1=12+222m+1+22+322m+1+12+3222m−1+12+4222m−1+22+2222m−1+32+4222m−132+322+22+422m+2−4=23/2+m5+1052m+2m+117−210+132m+1+4×21/2+m−217−85+22+10×2m−10,SOredD5=∑ijdi−12+dj−12=1−12+2−122m+1+2−12+3−122m+1+1−12+3−1222m−1+1−12+4−1222m−1+2−12+2−1222m−1+3−12+4−1222m−13−12+3−122+2−12+4−122m+2−4=6132m+52m+1+23/2+m+18×2m+22−613−14.


## 4. Conclusion

Topological indices found numerous applications in many regions of material science, arithmetic, informatics, biology, and so on. However, their most important use is in the nonexact quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR). Topological indices have an interconnection with the structure of the chemical structure. In this paper, we computed the newly introduced Sombor indices for phosphorus-containing dendrimers, porphyrin-cored dendrimers, PDI-cored dendrimers, triazine-based dendrimers, and aliphatic polyamide dendrimers. Figures7 and 8 give the graphical comparison of computed results for the aforementioned dendrimers.Figure 7
Comparison ofSO for D1m, D2m, D3m, D4m, and D5m.Figure 8
Comparison ofSOred for D1m, D2m, D3m, D4m, and D5m.



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*Source: 9930645-2021-05-07.xml*