# Investigation of Dendrimer Structures by Means of K-Banhatti Invariants

Authors: Cheng-Gang Huo; Fozia Azhar; Abaid Ur Rehman Virk; Tariq Ismaeel
Journal: Journal of Mathematics
 (2022)

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## Abstract

Dendrimers are highly branched macromolecules. The structural chemistry of dendrimers could be shaped by their topological invariants to target the particular design with appropriate properties to bring the drugs to mark a carrier vehicle. This study is about some new topological indices of dendrimer generations. Here, we calculate K Banhatti indices for five generations of dendrimers. Precisely speaking, we computed the 1st K Banhatti redefined Zagreb index, 2nd K Banhatti redefined Zagreb indices, and 3rd K Banhatti redefined Zagreb index for the dendrimersDip for i=1,2,3,4,5.


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## Body

## 1. Introduction

Dendrimers are highly structured, polymeric branching molecules. Similar dendrimer names include arborol and cascade molecules. In general, dendrimers are about the same in context and generally take on a three-dimensional circular morphology. The word dendron is also used interchangeably.The term dendrimers mean an artificially generated nanomolecular structure which is highly useful in drug delivery [1, 2]. Commonly, there are three types of dendrimers: (a) central core dendrimer, (b) interior surface dendrimer, and (c) outer surface dendrimers [3]. Due to their different molecular nanoshapes, organic science and dendrimes act as a solubilizing agent in different reactions [4]. For detailed applications of dendrimers, we recommend the readers [5, 6]. The structure of the dendrimer is well known and there are three main components of the dendrimer architecture;(i)
Initiator core(ii)
Interior layer(iii)
Terminal functionalitiesDue to their molecular architecture, dendrimers display some undoubtedly enhanced physical and chemical properties when correlated with a conventional liner polymer [2, 7]. The rare architectural layout of dendrimer, branching intensity, multivalency, globular, and well-defined molecular weight implies that dendrimers are rare and excellent nanocarriers in medical utilization such as drug delivery, gene transfection, tumor therapy, etc. The dendrimer architecture is shown in Figure 1.Figure 1
Dendrimer architecture.Consider a graphG with vertex set V and the edge set E. For a vertex v, the degree is denoted by dv which is the number of vertices at distance one from v. The distance between two vertices is the length of the shortest path between them.By a topological index, we mean a number associated with the graph that is unique up to graph isomorphism and that helps us to determine the hidden properties based on the symmetric structure of the graph [8–10]. Numerous methods are present in history to check the quality of a topological index. There are two main clashes of topological indices; first one is the degree-based topological indices and the second class is known as distance-based topological indices [11–13]. The study of topological index begun with the work of Wiener [14] in the year 1947 when he introduced the very first topological index while computing the boiling points of alkane which is today known as the Wiener index. The Wiener index is a distance-based topological index. In 1975, the very first degree-based topological index was introduced by Randić in [15] which is today known as the Randić index. After Randić index, many degree-based indices are introduced and studied by the researchers [16, 17]. All the introduced topological indices give some information about the concerned chemical structure [13]. For example, the redefined Zagreb indices are helpful to compute π-electronic energy of chemical structures [18].Continuing the work on topological indices, two variants of K Banhatti indices were put forward by Kulli in [19].(1)B1G=∑uedGu+dGe,B2G=∑uedGu×dGe.For other variants of K Banhatti indices, we recommend [20].Motivated by the work of Kulli [19], in this paper, we introduced the K Banhatti redefined Zagreb indices as follows:(2)KBR1G=∑uedu+dedu×de,KBR2G=∑uedu×dedu+de,KBR3G=∑uedu+dedu×de.In this report, we introduced a new class of topological indices for dendrimer generations. We calculated K Banhatti indices for five generations of dendrimers. Precisely speaking, we computed 1st K Banhatti redefined Zagreb index, 2nd K Banhatti redefined Zagreb indices, and 3rd K Banhatti redefined Zagreb index forD1ip, where i=1,2,3,4,5.


## 2. Main Results

In this section, we will discuss molecular graphs of five generations of dendrimers and calculate K Banhatti indices for abovementioned dendrimers; molecular graphs of these five generations can be found in Figures2–6.Figure 2
D1p for m = p.Figure 3
D2p for p = 1.Figure 4
D2p for p = 1.Figure 5
D4p for p = 2.Figure 6
D5p for m = p.


### 2.1. Phosphorus-Containing Dendrimers

The first generation of dendrimers we studied in this paper is phosphorus-containing dendrimers [2]. The graph of this generation is denoted by D1p where the number of stages is represented by p, see Figure 2.By observing Figure2, we can give the following remarks.Remark 1.
The graphD1p of phosphorus-containing dendrimers has 911×2p+1−8 vertices and 69×2p+2−13 edges.Remark 2.
The set of vertex of the graphD1p can be divided in to four different classes with respect to the degrees. The number of vertices of degree one are 42×2p−12, the number of vertices of degree two are 96×2p−39, the number of vertices of degree three 42×2p−18, and the number of vertices of degree four are 18×2p−3.Remark 3.
The edge set of the graphD1p can also be divided into different classes with respect to the degrees of the end vertices of edges and this division is given in Table 1.
Now, we are ready to present the first main result of this paper.Table 1
Partition of the edges ofD1p.
du,dvdeFrequency1,3262p−11,4365×2p−12,22182p+1−12,3362p+4−72,443×2p+33,4563×2p−1Theorem 1.
TheKBR1, KBR2, and KBR3 indices for D1p are as follows:(i)
KBR1D1p=293/5.2p−366/5+18.2p+1+5.2p+4+9/4.2p+3,(ii)
KBR2D1p=241/4.2p−1763/20+18.2p+1+36/5.2p+4+4.2p+3,(iii)
KBR3D1p=2556.2p−2376+288.2p+1+180.2p+4+144.2p+3.Proof.
(3)KBR1D1=∑uedu+dedu×de=1+21×26×2p−6+1+31×330×2p−6+2+22×218×2p+1−18+2+32×36×2p+4−42+2+42×43×2p+3+3+43×418×2p−6=2935.2p−3665+18.2p+1+5.2p+4+94.2p+3,KBR2D1=∑uedu×dedu+de=1×21+26×2p−6+1×31+330×2p−6+2×22+218×2p+1−18+2×32+36×2p+4−42+2×42+43×2p+3+3×43+418×2p−6=2414.2p−176320+18.2p+1+365.2p+4+4.2p+3,KBR3D1=∑uedu×dedu+de=1×21+26×2p−6+1×31+330×2p−6+2×22+218×2p+1−18+2×32+36×2p+4−42+2×42+43×2p+3+3×43+418×2p−6=2556.2p−2376+288.2p+1+180.2p+4+144.2p+3.


### 2.2. Porphyrin-Cored Dendrimer

Now, we study the porphyrin-cored dendrimer. The graph of porphyrin-cored dendrimer is denoted byD2p [2], see Figure 3.By observing Figure3, we can give the following remarks:Remark 4.
The graph of porphyrin-cored dendrimerD2p has 42p+3+9 vertices and 42p+3+11 edges.Remark 5.
The vertex set of porphyrin-cored dendrimerD2p can be divided into the four classes with respect to the degrees. The number of vertices of degree one is 12×2p−8, the number of vertices of degree two is 12×2p+32, the number of vertices of degree three are 4×2p+16, and the number of vertices of degree is 4×2p−4.Remark 6.
The edge set of porphyrin-cored dendrimerD2p can also be divided into different classes based on the degree of end vertices and this division is presented in Table 2.Table 2
Partition of the edges ofD2p.
du,dvdeFrequency1,214×2p1,324×2p−41,434×2p−42,224×2p+202,334×2p+322,448×2p−83,34123,454×2p−4Theorem 2.
For the porphyrin-cored dendrimerD2p, we have the following:(i)
KBR1D2p=487/15.2p+548/15(ii)
KBR2D2p=1109/30.2p+11089/210(iii)
KBR3D2p=1128.2p+1352Proof.
(4)KBR1D2=∑uedu+dedu×de=1+11×14×2p+1+21×24×2p−4+1+31×34×2p−4+2+22×24×2p+20+2+32×34×2p+32+2+42×48×2p−8+3+43×412+3+53×54×2p−4=48715.2p+54815,KBR2D2=∑uedu×dedu+de=1×11+14×2p+1×21+24×2p−4+1×31+34×2p−4+2×22+24×2p+20+2×32+34×2p+32+2×42+48×2p−8+3×43+412+3×53+54×2p−4=110930.2p+11089210,KBR3D2=∑uedu×dedu+de=1×11+14×2p+1×21+24×2p−4+1×31+34×2p−4+2×22+24×2p+20+2×32+34×2p+32+2×42+48×2p−8+3×43+412+3×53+54×2p−4=1128.2p+1352.


### 2.3. PDI-Cored Dendrimers

The graph of PDI-cored dendrimers is denoted byD3p [2], see Figure 4.By observing Figure4, we can give the following remarks:Remark 7.
The graph of PDI-cored dendrimersD3p has 20×2p+20 vertices and 20×22p+20 edges.Remark 8.
The vertex set of the graph of PDI-cored dendrimersD3p can be divided into different classes with respect to the degree of vertices. The number of vertices of degree one is 2×2p+1+4, the number of vertices of degree two is 6×2p+1, and the number of vertices of degree three is 2×2p+1+16.Remark 9.
The edge set of the graph of PDI-cored dendrimersD3p can also be divided into different classes with respect to the degree of the end vertices and this division is presented in Table 3.Table 3
Partition of the edges ofD3p.
du,dvdeFrequency1,212p+11,3242p−1+12,223×2p+1+12,3320×2p−13,3422Theorem 3.
TheKBR1, KBR2, and KBR3 indices for D3p are as follows:(i)
KBR1D3p=5.2p+1+68/3.2p−1+119/6(ii)
KBR2D3p=5.2p+1+80/3.2p−1+869/21(iii)
KBR3D3p=50.2p+1+624.2p−1+1888Proof.
(5)KBR1D3=∑uedu+dedu×de=1+11×12p+1+1+21×24×2p−1+4+3+33×322+2+22×23×2p+1+1+2+32×320×2p−1=5.2p+1+683.2p−1+1196,KBR2D3=∑uedu×dedu+de=1×11+12p+1+1×21+24×2p−1+4+3×33+322+2×22+23×2p+1+1+2×32+320×2p−1=5.2p+1+803.2p−1+86921,KBR3D3=∑uedu×dedu+de=1×11+12p+1+1×21+24×2p−1+4+3×33+322+2×22+23×2p+1+1+2×32+320×2p−1=50.2p+1+624.2p−1+1888.


### 2.4. Triazine-Based Dendrimer

The graph of triazine-based dendrimers is denoted byD4p [2], see Figure 5.By observing Figure5, we can give the following remarks:Remark 10.
The graph of the triazine-based dendrimerD4p has 25×22p+2+1/3 vertices and 7×22p+1+1 edges.Remark 11.
The vertices of the graph of triazine-based dendrimerD4p is divided into three classes. The number of vertices of degree one is 22p+1, the number of vertices of degree two is 22p+1+7×4p+1/6+4p+1/3, and the number of vertices of degree three is 4+5×4p+1/6−10/3.Remark 12.
The edges of the graph of triazine-based dendrimerD4p can also be divided into different classes and this division is presented in Table 4.Table 4
Partition of the edges ofD4p.
du,dvdeFrequency1,212p+12,2225×22p−2/32,33211×22p+4/33,342×22p+1−1/3Theorem 4.
TheKBR1, KBR2, and KBR3 indices for D4p are as follows:(i)
KBR1D4p=2.2p+1+85/9.22p+25/36+7/18.22p+1(ii)
KBR2D4p=2.2p+1+182/15.22p+136/105+8/7.22p+1(iii)
KBR3D4p=2.2p+1+820/3.22p+92/3+56.22p+1Proof.
(6)KBR1D4=∑uedu+dedu×de=1+21×22p+1+2+22×225×22p−23=+2+32×3211×22p+43+3+43×42×22p+1−13=2.2p+1+859.22p+2536+718.22p+1,KBR2D4=∑uedu×dedu+de=1×21+22p+1+2×22+225×22p−23+2×32+3211×22p+43+3×43+42×22p+1−13=2.2p+1+18215.22p+136105+87.22p+1,KBR3D4=∑uedu×dedu+de=1×21+22p+1+2×22+225×22p−23+2×32+3211×22p+43+3×43+42×22p+1−13=2.2p+1+8203.22p+923+56.22p+1.


### 2.5. Aliphatic Polyamide Dendrimers

The graph of aliphatic polyamide dendrimers is denoted byD5p [2], see Figure 6.By observing Figure6, we can give the following remarks:Remark 13.
The vertices of the graph of aliphatic polyamide dendrimersD5p can be divided into classes. The number of vertices of degree one is 43×2p−1−1, the number of vertices of degree two is 43×2p−1−1, and the number of vertices of degree three is 2p+122p−1.Remark 14.
The edges of the graph of aliphatic polyamide dendrimersD5p can also be divided into classes and this division is given in Table 5.Table 5
Partition of the edges ofD5p.
du,dvdeFrequency1,212p+12,332p+11,3222p−11,4322p−12,2222p−13,4522p−13,3422,442p+2−4Theorem 5.
TheKBR1, KBR2, and KBR3 indices for D5p are as follows:(i)
KBR1D5p=4/3.2p+1+106/15.2p−89/10+3/4.22p+2(ii)
KBR2D5p=16/5.2p+1+82/8.2p−1021/84+4/3.22p+2(iii)
KBR3D5p=32.2p+1+308.2p−332+48.22p+2Proof.
(7)KBR1D5=∑uedu+dedu×de+1+31×32×2p−2+1+41×42×2p−2+3+33×3d2+3+53×52×2p−2+2+42×42p+2−4=43.2p+1+10615.2p−8910+34.22p+2,KBR2D5=∑uedu×dedu+de+1×31+32×2p−2+1×41+42×2p−2+3×33+32+3×53+52×2p−2+2×42+42p+2−4=165.2p+1+828.2p−102184+43.22p+2,KBR3D5=∑uedu×dedu+de=1×21+22p+1+2×32+32p+1+1×21+22×2p−2+1×31+32×2p−2+1×41+42×2p−2+3×53+52×2p−2+3×33+32+2×42+42m+2−4=32.2p+1+308.2p−332+48.22p+2.


## 2.1. Phosphorus-Containing Dendrimers

The first generation of dendrimers we studied in this paper is phosphorus-containing dendrimers [2]. The graph of this generation is denoted by D1p where the number of stages is represented by p, see Figure 2.By observing Figure2, we can give the following remarks.Remark 1.
The graphD1p of phosphorus-containing dendrimers has 911×2p+1−8 vertices and 69×2p+2−13 edges.Remark 2.
The set of vertex of the graphD1p can be divided in to four different classes with respect to the degrees. The number of vertices of degree one are 42×2p−12, the number of vertices of degree two are 96×2p−39, the number of vertices of degree three 42×2p−18, and the number of vertices of degree four are 18×2p−3.Remark 3.
The edge set of the graphD1p can also be divided into different classes with respect to the degrees of the end vertices of edges and this division is given in Table 1.
Now, we are ready to present the first main result of this paper.Table 1
Partition of the edges ofD1p.
du,dvdeFrequency1,3262p−11,4365×2p−12,22182p+1−12,3362p+4−72,443×2p+33,4563×2p−1Theorem 1.
TheKBR1, KBR2, and KBR3 indices for D1p are as follows:(i)
KBR1D1p=293/5.2p−366/5+18.2p+1+5.2p+4+9/4.2p+3,(ii)
KBR2D1p=241/4.2p−1763/20+18.2p+1+36/5.2p+4+4.2p+3,(iii)
KBR3D1p=2556.2p−2376+288.2p+1+180.2p+4+144.2p+3.Proof.
(3)KBR1D1=∑uedu+dedu×de=1+21×26×2p−6+1+31×330×2p−6+2+22×218×2p+1−18+2+32×36×2p+4−42+2+42×43×2p+3+3+43×418×2p−6=2935.2p−3665+18.2p+1+5.2p+4+94.2p+3,KBR2D1=∑uedu×dedu+de=1×21+26×2p−6+1×31+330×2p−6+2×22+218×2p+1−18+2×32+36×2p+4−42+2×42+43×2p+3+3×43+418×2p−6=2414.2p−176320+18.2p+1+365.2p+4+4.2p+3,KBR3D1=∑uedu×dedu+de=1×21+26×2p−6+1×31+330×2p−6+2×22+218×2p+1−18+2×32+36×2p+4−42+2×42+43×2p+3+3×43+418×2p−6=2556.2p−2376+288.2p+1+180.2p+4+144.2p+3.


## 2.2. Porphyrin-Cored Dendrimer

Now, we study the porphyrin-cored dendrimer. The graph of porphyrin-cored dendrimer is denoted byD2p [2], see Figure 3.By observing Figure3, we can give the following remarks:Remark 4.
The graph of porphyrin-cored dendrimerD2p has 42p+3+9 vertices and 42p+3+11 edges.Remark 5.
The vertex set of porphyrin-cored dendrimerD2p can be divided into the four classes with respect to the degrees. The number of vertices of degree one is 12×2p−8, the number of vertices of degree two is 12×2p+32, the number of vertices of degree three are 4×2p+16, and the number of vertices of degree is 4×2p−4.Remark 6.
The edge set of porphyrin-cored dendrimerD2p can also be divided into different classes based on the degree of end vertices and this division is presented in Table 2.Table 2
Partition of the edges ofD2p.
du,dvdeFrequency1,214×2p1,324×2p−41,434×2p−42,224×2p+202,334×2p+322,448×2p−83,34123,454×2p−4Theorem 2.
For the porphyrin-cored dendrimerD2p, we have the following:(i)
KBR1D2p=487/15.2p+548/15(ii)
KBR2D2p=1109/30.2p+11089/210(iii)
KBR3D2p=1128.2p+1352Proof.
(4)KBR1D2=∑uedu+dedu×de=1+11×14×2p+1+21×24×2p−4+1+31×34×2p−4+2+22×24×2p+20+2+32×34×2p+32+2+42×48×2p−8+3+43×412+3+53×54×2p−4=48715.2p+54815,KBR2D2=∑uedu×dedu+de=1×11+14×2p+1×21+24×2p−4+1×31+34×2p−4+2×22+24×2p+20+2×32+34×2p+32+2×42+48×2p−8+3×43+412+3×53+54×2p−4=110930.2p+11089210,KBR3D2=∑uedu×dedu+de=1×11+14×2p+1×21+24×2p−4+1×31+34×2p−4+2×22+24×2p+20+2×32+34×2p+32+2×42+48×2p−8+3×43+412+3×53+54×2p−4=1128.2p+1352.


## 2.3. PDI-Cored Dendrimers

The graph of PDI-cored dendrimers is denoted byD3p [2], see Figure 4.By observing Figure4, we can give the following remarks:Remark 7.
The graph of PDI-cored dendrimersD3p has 20×2p+20 vertices and 20×22p+20 edges.Remark 8.
The vertex set of the graph of PDI-cored dendrimersD3p can be divided into different classes with respect to the degree of vertices. The number of vertices of degree one is 2×2p+1+4, the number of vertices of degree two is 6×2p+1, and the number of vertices of degree three is 2×2p+1+16.Remark 9.
The edge set of the graph of PDI-cored dendrimersD3p can also be divided into different classes with respect to the degree of the end vertices and this division is presented in Table 3.Table 3
Partition of the edges ofD3p.
du,dvdeFrequency1,212p+11,3242p−1+12,223×2p+1+12,3320×2p−13,3422Theorem 3.
TheKBR1, KBR2, and KBR3 indices for D3p are as follows:(i)
KBR1D3p=5.2p+1+68/3.2p−1+119/6(ii)
KBR2D3p=5.2p+1+80/3.2p−1+869/21(iii)
KBR3D3p=50.2p+1+624.2p−1+1888Proof.
(5)KBR1D3=∑uedu+dedu×de=1+11×12p+1+1+21×24×2p−1+4+3+33×322+2+22×23×2p+1+1+2+32×320×2p−1=5.2p+1+683.2p−1+1196,KBR2D3=∑uedu×dedu+de=1×11+12p+1+1×21+24×2p−1+4+3×33+322+2×22+23×2p+1+1+2×32+320×2p−1=5.2p+1+803.2p−1+86921,KBR3D3=∑uedu×dedu+de=1×11+12p+1+1×21+24×2p−1+4+3×33+322+2×22+23×2p+1+1+2×32+320×2p−1=50.2p+1+624.2p−1+1888.


## 2.4. Triazine-Based Dendrimer

The graph of triazine-based dendrimers is denoted byD4p [2], see Figure 5.By observing Figure5, we can give the following remarks:Remark 10.
The graph of the triazine-based dendrimerD4p has 25×22p+2+1/3 vertices and 7×22p+1+1 edges.Remark 11.
The vertices of the graph of triazine-based dendrimerD4p is divided into three classes. The number of vertices of degree one is 22p+1, the number of vertices of degree two is 22p+1+7×4p+1/6+4p+1/3, and the number of vertices of degree three is 4+5×4p+1/6−10/3.Remark 12.
The edges of the graph of triazine-based dendrimerD4p can also be divided into different classes and this division is presented in Table 4.Table 4
Partition of the edges ofD4p.
du,dvdeFrequency1,212p+12,2225×22p−2/32,33211×22p+4/33,342×22p+1−1/3Theorem 4.
TheKBR1, KBR2, and KBR3 indices for D4p are as follows:(i)
KBR1D4p=2.2p+1+85/9.22p+25/36+7/18.22p+1(ii)
KBR2D4p=2.2p+1+182/15.22p+136/105+8/7.22p+1(iii)
KBR3D4p=2.2p+1+820/3.22p+92/3+56.22p+1Proof.
(6)KBR1D4=∑uedu+dedu×de=1+21×22p+1+2+22×225×22p−23=+2+32×3211×22p+43+3+43×42×22p+1−13=2.2p+1+859.22p+2536+718.22p+1,KBR2D4=∑uedu×dedu+de=1×21+22p+1+2×22+225×22p−23+2×32+3211×22p+43+3×43+42×22p+1−13=2.2p+1+18215.22p+136105+87.22p+1,KBR3D4=∑uedu×dedu+de=1×21+22p+1+2×22+225×22p−23+2×32+3211×22p+43+3×43+42×22p+1−13=2.2p+1+8203.22p+923+56.22p+1.


## 2.5. Aliphatic Polyamide Dendrimers

The graph of aliphatic polyamide dendrimers is denoted byD5p [2], see Figure 6.By observing Figure6, we can give the following remarks:Remark 13.
The vertices of the graph of aliphatic polyamide dendrimersD5p can be divided into classes. The number of vertices of degree one is 43×2p−1−1, the number of vertices of degree two is 43×2p−1−1, and the number of vertices of degree three is 2p+122p−1.Remark 14.
The edges of the graph of aliphatic polyamide dendrimersD5p can also be divided into classes and this division is given in Table 5.Table 5
Partition of the edges ofD5p.
du,dvdeFrequency1,212p+12,332p+11,3222p−11,4322p−12,2222p−13,4522p−13,3422,442p+2−4Theorem 5.
TheKBR1, KBR2, and KBR3 indices for D5p are as follows:(i)
KBR1D5p=4/3.2p+1+106/15.2p−89/10+3/4.22p+2(ii)
KBR2D5p=16/5.2p+1+82/8.2p−1021/84+4/3.22p+2(iii)
KBR3D5p=32.2p+1+308.2p−332+48.22p+2Proof.
(7)KBR1D5=∑uedu+dedu×de+1+31×32×2p−2+1+41×42×2p−2+3+33×3d2+3+53×52×2p−2+2+42×42p+2−4=43.2p+1+10615.2p−8910+34.22p+2,KBR2D5=∑uedu×dedu+de+1×31+32×2p−2+1×41+42×2p−2+3×33+32+3×53+52×2p−2+2×42+42p+2−4=165.2p+1+828.2p−102184+43.22p+2,KBR3D5=∑uedu×dedu+de=1×21+22p+1+2×32+32p+1+1×21+22×2p−2+1×31+32×2p−2+1×41+42×2p−2+3×53+52×2p−2+3×33+32+2×42+42m+2−4=32.2p+1+308.2p−332+48.22p+2.


## 3. Graphical Representation

## 4. Concluding Remarks

The topological index is very helpful in biology, material science, informatics, arithmetic, etc. The most valuable use of a topological index is in QSPR and QSAR. By mean of the topological index, we can assign a number to the graph of a dendrimer and this number help us to determine the hidden information in the symmetric structure of dendrimer. This paper introduced and computed K Banhatti redefined Zagreb indices, i.e., 1st K Banhatti redefined Zagreb index, 2nd K Banhatti redefined Zagreb indices, and 3rd K Banhatti redefined Zagreb index for the dendrimersDip, where i=1,2,3,4,5. Figures 7, 8, and 9 show the graphical behavior of computed results of Dip, where i=1,2,3,4,5.Figure 7
Comparison ofKBR1 for Dip, where i=1,2,3,4,5.Figure 8
Comparison ofKBR1 for Dip, where i=1,2,3,4,5.Figure 9
Comparison ofKBR1 for Dip, where i=1,2,3,4,5.


## 5. Future Directions

It will be interesting to compute the different versions of graph energies of understudied dendrimers. It will also be interesting to define the corresponding energies for the presented topological indices.



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*Source: 4451899-2022-06-24.xml*