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# Nanotechnology Reviews

Editor-in-Chief: Hui, David

Managing Editor: Skoryna, Juliusz

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# Topological aspects of some dendrimer structures

• Corresponding author
• Department of Natural Sciences and Humanities, University of Engineering and Technology, Lahore (RCET), 54000, Pakistan
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• Other articles by this author:
• Department of Mathematics, Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, Lahore, 54000, Pakistan
• Other articles by this author:
/ Wei Gao
• School of Information Science and Technology, Yunnan Normal University, Kunming, 650500, China
• Other articles by this author:
/ Waqas Nazeer
Published Online: 2018-02-20 | DOI: https://doi.org/10.1515/ntrev-2017-0184

## Abstract

A numerical number associated to the molecular graph G that describes its molecular topology is called topological index. In the study of QSAR and QSPR, topological indices such as atom-bond connectivity index, Randić connectivity index, geometric index, etc. help to predict many physico-chemical properties of the chemical compound under study. Dendrimers are macromolecules and have many applications in chemistry, especially in self-assembly procedures and host-guest reactions. The aim of this report is to compute degree-based topological indices, namely the fourth atom-bond connectivity index and fifth geometric arithmetic index of poly propyl ether imine, zinc porphyrin, and porphyrin dendrimers.

## 1 Introduction

The branch of mathematics, precisely mathematical chemistry, in which graph theory is applied to solve molecular problems is called chemical graph theory (CGT). In CGT, a molecule is represented by a graph (molecular graph) in which atoms are taken as vertices and atomic bounds are taken as edges. The main objective of CGT is to associate a numerical number to the molecular graph, which correlates the chemical and physical properties of the molecule.

Dendrimers are repetitively branched molecules. A dendrimer is typically symmetric around the core, and often adopts a spherical three-dimensional morphology. The first dendrimers were made by divergent synthesis approaches by Vögtle in 1978 [1]. This molecule has wide applications in nanosciences, biology, and chemistry. A lot of research papers on computation of topological indices of dendrimers have been published [2], [3], [4], [5], [6].

A molecular graph G=(V,E) is a simple graph with the vertex set V(G) and the edge set E(G). The vertices represent nonhydrogen atoms and the edges represents covalent bonds between the corresponding atoms. The set of all the vertices adjacent to the vertex v is called the neighborhood, NG(v), of the vertex v. In graph G, d(v)=|NG(v)| represents the degree of v and $S\left(v\right)={\sum }_{u\in {N}_{G}\left(u\right)}d\left(u\right).$ We used the standard notations from graph theory mostly taken from books [7], [8].

In 1975, Randić [9] proposed the first degree based topological index and defined as $R\left(G\right)=\frac{1}{d\left(u\right)d\left(v\right)}.$ After that, Gutman and Tirnajstić [10] introduced the first Zagreb index and the second Zagreb index for a simply connected graph. Estrada et al. [11] defined the well-known topological index named as atom-bond connectivity (ABC) index. This topological index can be used to model the enthalpy of formation of alkanes and is defined as:

$ABC(G)=∑uv∈E(G)d(u)+d(v)−2d(u)d(v).$(1)

The fourth version of ABC index was proposed by Ghorbani and Hosseinzadeh [12]:

$ABC4(G)=∑uv∈E(G)S(u)+S(v)−2S(u)S(v).$(2)

Vukičević and Furtula [13] introduced another well-known topological index called geometric-arithmetic (GA) index. The predictive power of the GA index is better than ${R}_{-\text{\hspace{0.17em}}\frac{1}{2}}$ for many physicochemical properties, for example entropy, boiling point, vaporization, enthalpy, acentric factor, and enthalpy of formation. For a simply connected graph G, the GA index is defined as:

$GA(G)=∑uv∈E(G)2d(u)d(v)d(u)+d(v).$(3)

Graovac et al. [14] introduced the fifth version of GA index and defined as:

$GA5(G)=∑uv∈E(G)2S(u)S(v)S(u)+S(v).$(4)

For further history and results on topological indices, see [12], [15], [16], [17], [18], [19], [20], [21], [22].

In this paper, we compute the fourth ABC (ABC4) and fifth GA (GA5) indices for porphyrin DnPn, zinc porphyrin DPZn and poly propyl ether imine (PETIM) dendrimer structure.

## 2 Discussion and main results

This section contains computational results of some dendrimers. We give close formulae of GA5 and ABC4 indices of some dendrimers. We start with the PETIM dendrimer of generation having growth stages n. Figure 1 shows the growth of PETIM dendrimer. In the molecular structure of PETIM dendrimer, there are 24×2n−23 atoms and 24×2n−24 bonds.

Figure 1:

The molecular structure of PETIM dendrimer with nth growth.

In the following result, we computed the ABC4 index of PETIM dendrimer.

Theorem 1. Let G be the molecular graph of PETIM dendrimer, then the ABC4 index of G is

$AB{C}_{4}\left(G\right)=14.44×{2}^{n}-3×7.6117.$

Proof. For the molecular graph G of PETIM dendrimer, |V(G)|=24×2n−23 and |E(G)|=24×2n−24. We divide the edge set based on the sum of degree of neighbors of end-vertices of every edge. This partition is shown in Table 1.

Table 1:

Edge set partition of PETIM dendrimer based on sum of degrees of neighbors of end-vertices of every edge.

Now from Table 1,

$\begin{array}{c}AB{C}_{4}\left(PETIM\right)=\sum _{uv\in E\left(PETIM\right)}\sqrt{\frac{S\left(u\right)+S\left(v\right)-2}{S\left(u\right)S\left(v\right)}}\\ =2×{2}^{n}\sqrt{\frac{2+3-2}{2×3}}+2×{2}^{n}\sqrt{\frac{3+4-2}{3×4}}\\ +\left(8×{2}^{n}-12\right)\sqrt{\frac{4+4-2}{4×4}}+\left(6×{2}^{n}-6\right)\sqrt{\frac{4+5-2}{4×5}}\\ +\left(6×{2}^{n}-6\right)\sqrt{\frac{5+6-2}{5×6}}\end{array}$

After some simplification, we obtain

$AB{C}_{4}\left(G\right)=14.44×{2}^{n}-3×7.6117.$

The following result gives the GA5 index of PETIM dendrimer.

Theorem 2. Let G be the molecular graph of PETIM dendrimer, then the GA5 index of G is

$G{A}_{5}\left(G\right)=2.9846×{2}^{n+3}-23.938.$

Proof. For the molecular graph G of PETIM dendrimer. From the definition of GA5, we have

$\begin{array}{c}G{A}_{5}\left(G\right)=\sum _{uv\in E\left(PETIM\right)}\frac{2\sqrt{S\left(u\right)S\left(v\right)}}{S\left(u\right)+S\left(v\right)}\\ =2×{2}^{n}\frac{2\sqrt{2×3}}{2+3}+2×{2}^{n}\frac{2\sqrt{3×4}}{3+4}+\left(8×{2}^{n}-12\right)\frac{2\sqrt{4×4}}{4+4}\\ +\left(6×{2}^{n}-6\right)\frac{2\sqrt{4×5}}{4+5}+\left(6×{2}^{n}-6\right)\frac{2\sqrt{5×6}}{5+6}.\end{array}$

After some simplification, we get

$G{A}_{5}\left(G\right)=2.9846×{2}^{n+3}-23.938.$

Now, we compute the ABC4 and GA5 indices of zinc porphyrin dendrimer DPZn. The molecular graph of DPZn has 96n−10 atoms and 105n−11 bonds. The molecular graph of zinc porphyrin dendrimer DPZn is shown in Figure 2. In Table 2, the edge-set partition of DPZn is given.

Figure 2:

Molecular graph of zinc porphyrin dendrimer DPZn.

Table 2:

Edge set partition of zinc porphyrin dendrimer DPZn based on sum of degrees of vertices adjacent to end-vertices of every edge.

The following theorem is about the ABC4 index for zinc porphyrin dendrimer DPZn.

Theorem 3. Let DPZn be the molecular graph of zinc porphyrin dendrimer. Then

$AB{C}_{4}\left(DP{Z}_{n}\right)=4.9771×{2}^{n+3}-10.1770.$

Proof. Consider the graph of DPZn dendrimer. The graphical structure of DPZn contains V|DPZn|=56×2n−7 vertices and E|DPZn|=64×2n−4. We find the edge set partition based on the sum of degrees of neighbors of end-vertices of every edge for the zinc porphyrin dendrimer. Table 2 shows the edge partition of DPZn depending upon sum of degree of neighbors of end-vertices of every edge. Now, by using definition and Table 2, we have

$AB{C}_{4}\left(G\right)=\sum _{uv\in E\left(G\right)}\sqrt{\frac{S\left(u\right)+S\left(v\right)-2}{S\left(u\right)S\left(v\right)}}.$

So,

$\begin{array}{l}AB{C}_{4}\left(DP{Z}_{n}\right)=\sum _{uv\in E\left(DP{Z}_{n}\right)}\sqrt{\frac{S\left(u\right)+S\left(v\right)-2}{S\left(u\right)S\left(v\right)}}\\ \text{ }=8×{2}^{n}\sqrt{\frac{4+4-2}{4×4}}+8\sqrt{\frac{4+5-2}{4×5}}\\ \text{ }+\left(16×{2}^{n}-12\right)\sqrt{\frac{5+5-2}{5×5}}+\left(8×{2}^{n}-12\right)\sqrt{\frac{5+6-2}{5×6}}\\ \text{ }+\left(16×{2}^{n}-12\right)\sqrt{\frac{5+7-2}{5×7}}+8\sqrt{\frac{5+8-2}{5×8}}\\ \text{ }+\left(4×{2}^{n}\right)\sqrt{\frac{6+6-2}{6×6}}+4\sqrt{\frac{6+7-2}{6×7}}\\ \text{ }+\left(4×{2}^{n}-4\right)\sqrt{\frac{6+8-2}{6×8}}+\left(8×{2}^{n}-8\right)\sqrt{\frac{7+8-2}{7×8}}\\ \text{ }+4\sqrt{\frac{7+9-2}{7×9}}+8\sqrt{\frac{8+9-2}{8×9}}+8\sqrt{\frac{8+10-2}{8×10}}\\ \text{ }+4\sqrt{\frac{10+12-2}{10×12}}.\end{array}$

After some simplification, we obtain

$AB{C}_{4}\left(DP{Z}_{n}\right)=4.9771×{2}^{n+3}-10.1770.$

The following theorem gives the information about the computation of the GA5 of porphyrin dendrimer DPZn.

Theorem 4. Let DPZn be the molecular graph of zinc porphyrin dendrimer. Then

$G{A}_{5}\left(DP{Z}_{n}\right)=3.9802×{2}^{n+4}-0.1135.$

Proof. From the definition and Table 2, we have

$\begin{array}{l}G{A}_{5}\left(DP{Z}_{n}\right)=\sum _{uv\in E\left(DP{Z}_{n}\right)}\frac{2\sqrt{S\left(u\right)S\left(v\right)}}{S\left(u\right)+S\left(v\right)}\\ \text{ }=8×{2}^{n}\frac{2\sqrt{4×4}}{4+4}+8\frac{2\sqrt{4×5}}{4+5}+\left(16×{2}^{n}-12\right)\frac{2\sqrt{5×5}}{5+5}\\ \text{ }+\left(8×{2}^{n}-12\right)\frac{2\sqrt{5×6}}{5+6}+\left(16×{2}^{n}-12\right)\frac{2\sqrt{5×7}}{5+7}\\ \text{ }+8\frac{2\sqrt{5×8}}{5+8}+\left(4×{2}^{n}\right)\frac{2\sqrt{6×6}}{6+6}+4\frac{2\sqrt{6×7}}{6+7}\\ \text{ }+\left(4×{2}^{n}-4\right)\frac{2\sqrt{6×8}}{6+8}+\left(8×{2}^{n}-8\right)\frac{2\sqrt{7×8}}{7+8}\\ \text{ }+4\frac{2\sqrt{7×9}}{7+9}+8\frac{2\sqrt{8×9}}{8+9}+8\frac{2\sqrt{8×10}}{8+10}\\ \text{ }+4\frac{2\sqrt{10×12}}{10+12}.\end{array}$

After some simplification, we obtain

$G{A}_{5}\left(DP{Z}_{n}\right)=3.9802×{2}^{n+4}-0.1135.$

Now, we shall compute the ABC4 and GA5 topological indices of porphyrin dendrimers DnPn for n>0. Note that n=2m, where m≥2. The number of vertices and the number of edge in the structure of porphyrin dendrimers are 96n− 10 and 105n−11, respectively. The n growth stages of porphyrin dendrimer is shown in Figure 3. Now, we compute the ABC4 and GA5 indices of porphyrin dendrimer DnPn.

Figure 3:

Molecular graph of porphyrin dendrimer DnPn.

Theorem 5. Consider the graph of porphyrin dendrimer DnPn. Then,

$AB{C}_{4}\left({D}_{n}{P}_{n}\right)=53.85n+4.1951$

Proof. Consider the graph of DnPn dendrimer. The molecular graph of DnPn dendrimer has V|DnPn|=96×2n−10 vertices and E|DnPn|=105×2n−11. The edge set partition of DnPn dendrimer based on the sum of degrees of neighbors of end-vertices of every edge is shown in Table 3. With the help of Table 3, we have

Table 3:

Edge set partition of porphyrin dendrimer DnPn based on sum of degrees of vertices adjacent to end vertices of every edge.

$AB{C}_{4}\left(G\right)=\sum _{uv\in E\left(G\right)}\sqrt{\frac{S\left(u\right)+S\left(v\right)-2}{S\left(u\right)S\left(v\right)}}.$

So,

$\begin{array}{l}AB{C}_{4}\left({D}_{n}{P}_{n}\right)=\sum _{uv\in E\left({D}_{n}{P}_{n}\right)}\sqrt{\frac{S\left(u\right)+S\left(v\right)-2}{S\left(u\right)S\left(v\right)}}\\ \text{ }=2n\sqrt{\frac{3+5-2}{3×5}}+\left(n+1\right)\sqrt{\frac{4+4-2}{4×4}}+\left(8n-6\right)\sqrt{\frac{4+5-2}{4×5}}\\ \text{ }+24n\sqrt{\frac{4+6-2}{4×6}}+4n\sqrt{\frac{5+5-2}{5×5}}\\ \text{ }+\left(4n-6\right)\sqrt{\frac{5+6-2}{5×6}}+6n\sqrt{\frac{5+7-2}{5×7}}+18n\sqrt{\frac{6+7-2}{6×7}}\\ \text{ }+25n\sqrt{\frac{6+8-2}{6×8}}+n\sqrt{\frac{7+8-2}{7×8}}\\ \text{ }+11n\sqrt{\frac{7+9-2}{7×9}}+n\sqrt{\frac{8+9-2}{8×9}}\\ \text{ }=53.85n+4.1951.\end{array}$

The following theorem tells about the closed formula of GA5 for porphyrin dendrimer DnPn.

Theorem 6. Consider the molecular graph of porphyrin dendrimer DnPn dendrimer. Then, the GA5 index of DnPn dendrimer is equal to

$G{A}_{5}\left({D}_{n}{P}_{n}\right)=103.9015n-10.938$

Proof. From the definition of ABC4(G) and Table 3, we have

$\begin{array}{c}G{A}_{5}\left({D}_{n}{P}_{n}\right)=\sum _{uv\in E\left({D}_{n}{P}_{n}\right)}\frac{2\sqrt{S\left(u\right)×S\left(v\right)}}{S\left(u\right)+S\left(v\right)}\\ =2n\frac{2\sqrt{3×5}}{3+5}+\left(n+1\right)\frac{2\sqrt{4×4}}{4+4}+\left(8n-6\right)\frac{2\sqrt{4×5}}{4+5}\\ +24n\frac{2\sqrt{4×6}}{4+6}+4n\frac{2\sqrt{5×5}}{5+5}\\ +\left(4n-6\right)\frac{2\sqrt{5×6}}{5+6}+6n\frac{2\sqrt{5×7}}{5+7}+18n\frac{2\sqrt{6×7}}{6+7}\\ +25n\frac{2\sqrt{6×8}}{6+8}+n\frac{2\sqrt{7×8}}{7+8}\\ +11n\frac{2\sqrt{7×9}}{7+9}+n\frac{2\sqrt{8×9}}{8+9}.\end{array}$

After some simplification, we obtain the result:

$G{A}_{5}\left({D}_{n}{P}_{n}\right)=103.9015n-10.938.$

## 3 Conclusion

In this paper, we dealt with zinc porphyrin, poly propyl ether imine, and zinc porphyrin dendrimer structures and studied their topological indices. We have computed the fourth version of ABC index and fifth version of the GA index for these families of dendrimers.

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Accepted: 2018-01-22

Published Online: 2018-02-20

Published in Print: 2018-04-25

Citation Information: Nanotechnology Reviews, Volume 7, Issue 2, Pages 123–129, ISSN (Online) 2191-9097, ISSN (Print) 2191-9089,

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