Some Reverse Degree-Based Topological Indices and Polynomials of Dendrimers

Abstract

Topological indices collect information from the graph of molecule and help to predict properties of the underlying molecule. Zagreb indices are among the most studied topological indices due to their applications in chemistry. In this paper, we compute first and second reverse Zagreb indices, reverse hyper-Zagreb indices and their polynomials of Prophyrin, Propyl ether imine, Zinc Porphyrin and Poly (ethylene amido amine) dendrimers.

1. Introduction

Dendrimers, from a Greek word that translates to “trees” [1,2], are repetitively branched molecules. Dendrimers are generally symmetrical about the core and generally adopt a spherical three-dimensional morphology. Word dendrites are also often encountered. Dendrites usually contain a single chemically addressable group called the focus or core. The first dendrimer was made by Fritz Vogtle [3] using different synthetic methods, such as by RG Denkewalter at Allied [4,5] and Donald Tomalia at Dow Chemical [6,7,8]. George R. Newkome, Craig Hawker and Jean Frechet in 1990 [9] introduced a fusion synthesis method. The popularity of dendrimers has greatly increased. By 2005, there were more than 5000 scientific papers and patents.

Uses of dendrimers include conjugating other chemical species to the dendrimer surface that can work as distinguishing operators, (for example, a dye molecule), targeting components, affinity ligands, imaging agents, radioligands or pharmaceutical compounds. Dendrimers have exceptionally solid potential for these applications as their structure can prompt multivalent systems. As such, one dendrimer particle has several conceivable sites to couple to an active species. Scientists expected to use the hydrophobic environments of the dendritic media to conduct photochemical reactions that create the items that are synthetically challenged. Carboxylic acid and phenol-terminated water–solvent dendrimers have been incorporated to set up their utility in tranquilizer conveyance, leading to compound responses in their insides. This may enable specialists to connect both focusing molecules and drug molecules to the same dendrimer, which could lessen negative symptoms of medications on healthy cells. Due to these applications, dendrimers are extensively studied [10,11,12,13,14,15,16]. Here, we study Prophyrin, Propyl ether imine, Zinc Porphyrin and Poly(ethylene amido amine) dendrimers (Figure 1, Figure 2, Figure 3 and Figure 4).

Aslam et al. [16] studied three New/Old vertex-degree-based topological indices of these dendrimers. Gao et al., in 2018, ref. [17] computed eccentricity-based topological indices of Porphyrin-cored dendrimers. In the same year, Kang et al. [18] computed eccentricity-based topological indices of phosphorus-containing dendrimers. Some other degree-based topological indices of these dendrimers have been computed [19]. Figure 1, Figure 2, Figure 3 and Figure 4 are taken from [17,18,19].

Topological indices correspond to certain physicochemical properties such as boiling point, stability, strain energy and so forth of a chemical compound. Currently, there are more than 148 topological indices and none of them completely describe all properties of the molecular compounds under study, so there is always room to define new topological indices. Our aim was to study Prophyrin, Propyl ether imine, Zinc Porphyrin and Poly(ethylene amido amine) dendrimers. We computed the first and second reverse Zagreb indices, reverse hyper-Zagreb indices and their polynomials of these dendrimers. Graphical comparison of our results is also presented.

2. Preliminaries

A graph having no loop or multiple edges is known as a simple graph. A molecular graph is a simple graph in which atoms and bonds are represented by vertices and edges, respectively. The degree of a vertex v is the number of edges attached with it and is denoted by $d_v$. The maximum degree of vertex among the vertices of a graph is denoted by $\Delta (G)$. Kulli [20] introduced the concept of reverse vertex degree $c_v$, as $c_v=\Delta (G)-d_v)+1$. Throughout this paper, G denotes the simple graph, E denotes the edge set of G, V denotes the vertex set of G and $\left|X\right|$ denotes the cardinality of any set X.

In discrete mathematics, graph theory is not only the study of different properties of objects but also tells us about objects having same properties as investigating object [21]. In particular, graph polynomials related to graph are rich in information [22,23,24,25,26,27]. Mathematical tools such as polynomials and topological based numbers have significant importance to collect information about properties of chemical compounds [28,29,30]. We can find out hidden information about compounds through theses tools. Multifold graph polynomials are present in the literature. Actually, topological indices are numeric quantities that tell us about the whole structure of graph. There are many topological indices [31,32,33,34] that help us to study physical, chemical reactivities and biological properties. Wiener [35], in 1947, firstly introduced the concept of topological index while working on boiling point. Hosoya polynomial [22] plays an important role in the area of distance-based topological indices; we can find Wiener index, Hyper Wiener index and Tratch–Stankevich–Zefirove index from Hosoya polynomial. Randić index defined by Milan Randić [36] in 1975 is one of the oldest degree based topological indices and has been extensively studied by mathematician and chemists [37,38,39,40,41]. Later, Gutman et al. introduced the first and second Zagreb indices as

$$M_1(G)=\sum _{uvϵE(G)}(d_u+d_v),$$

and

$$M_2(G)=\sum _{uvϵE(G)}(d_u.d_v),$$

respectively.

Zagreb indices help us in finding $\mathsf{\Pi }$ electronic energy [42]. Many papers [43,44,45,46,47,48], surveys [42,49] and many modification of Zagreb indices are presented in the literature [20,50,51,52,53,54]. First and second Zagreb polynomials were defined in [26] as:

$$M_1(G,x)=\sum _{uvϵE(G)}{x}^{(d_u+d_v)},$$

and

$$M_2(G,x)=\sum _{uvϵE(G)}{x}^{(d_u.d_v)},$$

respectively.

Shirdel et al. [55] proposed the first and second hyper-Zagreb indices as:

$$H_1(G)=\sum _{uvϵE(G)}{(d_u+d_v)}^2,$$

and

$$H_2(G)=\sum _{uvϵE(G)}{(d_u.d_v)}^2.$$

Motivated by these indices, the first and second reverse Zagreb indices was defined in [20] as:

$$C{M}_1(G)=\sum _{uvϵE(G)}(c_u+c_v),$$

and

$$C{M}_2(G)=\sum _{uvϵE(G)}(c_u.c_v).$$

The first and second Reverse hyper-Zagreb indices was also defined in the same paper as:

$$HC{M}_1(G)=\sum _{uvϵE(G)}{(c_u+c_v)}^2,$$

and

$$HC{M}_2(G)=\sum _{uvϵE(G)}{(c_u.c_v)}^2.$$

With the help of reverse Zagreb and hyper-Zagreb indices, we are now able to define the reverse Zagreb and reverse hyper-Zagreb polynomials. For a simply connected graph G, the first and second reverse Zagreb polynomials are defined as:

$$C{M}_1(G,x)=\sum _{uvϵE(G)}{x}^{(c_u+c_v)},$$

and

$$C{M}_2(G,x)=\sum _{uvϵE(G)}{x}^{(c_u.c_v)}$$

and the first and second hyper-Zagreb polynomials are defined as:

$$HC{M}_1(G,x)=\sum _{uvϵE(G)}{x}^{{(c_u+c_v)}^2},$$

and

$$HC{M}_2(G,x)=\sum _{uvϵE(G)}{x}^{{(c_u.c_v)}^2}.$$

3. Main Results

In this section, we compute reverse Zagreb and reverse hyper-Zagreb indices of Prophyrin, Propyl ether imine, Zinc Porphyrin and Poly(ethylene amido amine) dendrimers.

3.1. Prophyrin Dendrimer $D_n{P}_n$

Theorem 1.

Let $D_n{P}_n$ be a prophyrin Dendrimer. Then, the first and second reverse Zagreb indices are

1. 

$C{M}_1(D_n{P}_n)=508n-60.$

2. 

$C{M}_2(D_n{P}_n)=558n-81.$

Proof. 

In the Prophyrin dendrimer $D_n{P}_n$, there are $96n-10$ vertices and $105n-11$ edges. Based on the degree of end vertices, the edge set of $D_n{P}_n$ can be divided into following sic classes

$$E_1(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=1,d_v=3\},$$

$$E_2(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=1,d_v=4\},$$

$$E_3(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=2,d_v=2\},$$

$$E_4(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=2,d_v=3\},$$

$$E_5(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=3,d_v=3\},$$

$$E_6(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=3,d_v=4\}.$$

In Figure 1, one can count easily that $|E_1(D_n{P}_n)|=2n$, $|E_2(D_n{P}_n)|=24n$, $|E_3(D_n{P}_n)|=10n-5$, $|E_4(D_n{P}_n)|=48n-6$, $|E_5(D_n{P}_n)|=13n$ and $|E_6(D_n{P}_n)|=8n.$

The maximum vertex degree $\Delta (G)$ in $D_n{P}_n$ is 4, so we have following six types of reverse edges in $D_n{P}_n$.

$$C{E}_1(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=4,d_v=2\},$$

$$C{E}_2(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=4,d_v=1\},$$

$$C{E}_3(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=3,d_v=3\},$$

$$C{E}_4(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=3,d_v=2\},$$

$$C{E}_5(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=2,d_v=2\},$$

$$C{E}_6(D_n{P}_n)=\{uvϵE(D_n{P}_n);d_u=2,d_v=1\}.$$

In addition, $|C{E}_1(D_n{P}_n)|=2n$, $|C{E}_2(D_n{P}_n)|=24n$, $|C{E}_3(D_n{P}_n)|=10n-5$, $|C{E}_4(D_n{P}_n)|=48n-6$, $|C{E}_5(D_n{P}_n)|=13n$ and $|C{E}_6(D_n{P}_n)|=8n.$

(i)

Now, using the definition of reverse first Zagreb index, we have

$$\begin{array}{ccc}\hfill C{M}_1(D_n{P}_n)& =& \sum _{uvϵE(G)}(c_u+c_v)\hfill \\ & =& (4+2)(2n)+(4+1)(24n)+(3+3)(10n-5)\hfill \\ & & (3+2)(18n-6)+(2+2)(13n)+(2+1)(8n)\hfill \\ & =& 508n-60.\hfill \end{array}$$

(ii)

Using the definition of reverse second Zagreb index, we have

$$\begin{array}{ccc}\hfill C{M}_2(D_n{P}_n)& =& \sum _{uvϵE(G)}(c_u.c_v)\hfill \\ & =& (4.2)(2n)+(4.1)(24n)+(3.3)(10n-5)\hfill \\ & & +(3.2)(18n-6)+(2.2)(13n)+(2.1)(8n)\hfill \\ & =& 558n-81.\hfill \end{array}$$

 □

Theorem 2.

The first and second reverse Zagreb polynomials of $D_n{P}_n$ are

1. 

$C{M}_1(D_n{P}_n,x)=(12n-5)x^6+(72n-6)x^5+13n{x}^4+8n{x}^3.$

2. 

$C{M}_2(D_n{P}_n,x)=(10n-5)x^9+2n{x}^8+(48n-6)x^6+37n{x}^4+8n{x}^2.$

Proof. 

(i)

Using the information given in Theorem 1 and definition of reverse first Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill C{M}_1(D_n{P}_n,x)& =& \sum _{uvϵE(G)}{x}^{(c_u+c_v)}\hfill \\ & =& (2n)x^{(4+2)}+(24n)x^{(4+1)}+(10n-5)x^{(3+3)}\hfill \\ & & +(18n-6)x^{(3+2)}+(13n)x^{(2+2)}+(8n)x^{(2+1)}\hfill \\ & =& (12n-5)x^6+(72n-6)x^5+13n{x}^4+8n{x}^3.\hfill \end{array}$$

(ii)

Using the information given in Theorem 1 and definition of reverse second Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill C{M}_2(D_n{P}_n,x)& =& \sum _{uvϵE(G)}{x}^{(c_u.c_v)}\hfill \\ & =& (2n)x^{(4.2)}+(24n)x^{(4.1)}+(10n-5)x^{(3.3)}\hfill \\ & & +(18n-6)x^{(3.2)}+(13n)x^{(2.2)}+(8n)x^{(2.1)}\hfill \\ & =& (10n-5)x^9+2n{x}^8+(48n-6)x^6+37n{x}^4+8n{x}^2.\hfill \end{array}$$

 □

Theorem 3.

The first and second reverse hyper-Zagreb indices of prophyrin Dendrimer $D_n{P}_n$ are

1. 

$HC{M}_1(D_n{P}_n)=2512n-330.$

2. 

$HC{M}_2(D_n{P}_n)=3290n-621.$

Proof. 

(i)

Using the information given in Theorem 1 and definition of reverse first hyper-Zagreb index, we have

$$\begin{array}{ccc}\hfill HC{M}_1(D_n{P}_n)& =& \sum _{uvϵE(G)}{(c_u+c_v)}^2\hfill \\ & =& {(4+2)}^2(2n)+{(4+1)}^2(24n)+{(3+3)}^2(10n-5)\hfill \\ & & +{(3+2)}^2(18n-6)+{(2+2)}^2(13n)+{(2+1)}^2(8n)\hfill \\ & =& 2512n-330.\hfill \end{array}$$

(ii)

Using the information given in Theorem 1 and definition of reverse second hyper-Zagreb index, we have

$$\begin{array}{ccc}\hfill HC{M}_2(D_n{P}_n)& =& \sum _{uvϵE(G)}{(c_u.c_v)}^2\hfill \\ & =& {(4.2)}^2(2n)+{(4.1)}^2(24n)+{(3.3)}^2(10n-5)\hfill \\ & & +{(3.2)}^2(18n-6)+{(2.2)}^2(13n)+{(2.1)}^2(8n)\hfill \\ & =& 3290n-621\hfill \end{array}$$

 □

Theorem 4.

The first and second reverse hyper-Zagreb polynomials of $D_n{P}_n$ are

1. 

$HC{M}_1(D_n{P}_n,x)=(12n-5)x^{36}+(72n-6)x^{25}+13n{x}^{16}+8n{x}^9.$

2. 

$HC{M}_1(D_n{P}_n,x)=(10n-5)x^{81}+2n{x}^{64}+(48n-6)x^{36}+37n{x}^{16}+8n{x}^4.$

Proof. 

(i)

Using the information given in Theorem 1 and definition of reverse first hyper-Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill C{M}_1(D_n{P}_n,x)& =& \sum _{uvϵE(G)}{x}^{{(c_u+c_v)}^2}\hfill \\ & =& (2n)x^{{(4+2)}^2}+(24n)x^{{(4+1)}^2}+(10n-5)x^{{(3+3)}^2}\hfill \\ & & +(18n-6)x^{{(3+2)}^2}+(13n)x^{{(2+2)}^2}+(8n)x^{{(2+1)}^2}\hfill \\ & =& (12n-5)x^{36}+(72n-6)x^{25}+13n{x}^{16}+8n{x}^9.\hfill \end{array}$$

(ii)

Using the information given in Theorem 1 and definition of reverse second hyper-Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill C{M}_2(D_n{P}_n,x)& =& \sum _{uvϵE(G)}{x}^{{(c_u.c_v)}^2}\hfill \\ & =& (2n)x^{{(4.2)}^2}+(24n)x^{{(4.1)}^2}+(10n-5)x^{{(3.3)}^2}\hfill \\ & & +(18n-6)x^{{(3.2)}^2}+(13n)x^{{(2.2)}^2}+(8n)x^{{(2.1)}^2}\hfill \\ & =& (10n-5)x^{81}+2n{x}^{64}+(48n-6)x^{36}+37n{x}^4+8n{x}^4.\hfill \end{array}$$

 □

The values of first and second reverse Zagreb indices and first and second reverse hyper-Zagreb indices of $D_n{P}_n$ for specific values of n are given in

Table 1. Topological indices of $D_n{P}_n$.

	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
First Reverse Zagreb Index	448	956	1464	1972	2480	2988	3496	4004	4512
Second Reverse Zagreb Index	497	1055	1613	2171	2729	3287	3845	4403	4961
First Reverse Hyper-Zagreb Index	2182	4694	7206	9718	12,230	14,742	17,254	19,766	22,278
Second Reverse Hyper-Zagreb Index	2669	5959	9249	12,539	15,829	19,119	22,409	25,699	28,989

.

3.2. Propyl Ether Imine Dendrimer $(PETIM)$

In this section, we compute reverse Zagreb indices, reverse Zagreb polynomials, reverse hyper-Zagreb indices and reverse hyper-Zagreb polynomials of Propyl Ether Imine dendrimer $PETIM$.

Theorem 5.

The first and second reverse Zagreb indices of $PETIM$ are

1. 

$C{M}_1(PETIM)={4.2}^{n+4}+{3.2}^{n+1}+{3.2}^n-102.$

2. 

$C{M}_2(PETIM)={4.2}^{n+4}+{2.2}^{n+1}+{36.2}^n-108.$

Proof. 

In Propyl Ether Imine dendrimer $PETIM$, there are ${24.2}^n-23$ vertices and ${24.2}^n-24$ edges. Based on the degree of end vertices, the edge set of $PETIM$ can be divided into following three classes.

$$E_1(PETIM)=\{uvϵE(PETIM);d_u=1,d_v=2\},$$

$$E_2(PETIM)=\{uvϵE(PETIM);d_u=2,d_v=2\},$$

$$E_3(PETIM)=\{uvϵE(PETIM);d_u=2,d_v=3\}.$$

In Figure 2, one can count easily that $|E_1(PETIM)|=2^{n+1}$, $|E_1(PETIM)|=2^{n+4}-18$ and $|E_1(PETIM)|={6.2}^n-6$.

The maximum vertex degree $\Delta (G)$ of $PETIM$ is 3, so we have following types of reverse edges.

$$C{E}_1(PETIM)=\{uvϵCE(PETIM);c_u=3,c_v=2\}.$$

$$C{E}_2(PETIM)=\{uvϵCE(PETIM);c_u=2,c_v=2\}.$$

$$C{E}_3(PETIM)=\{uvϵCE(PETIM);c_u=2,c_v=1\}.$$

Obviously, we have $|C{E}_1(PETIM)|=2^{n+1}$, $|C{E}_1(PETIM)|=2^{n+4}-18$ and $|C{E}_1(PETIM)|={6.2}^n-6$.

(i)

Now, from the definition of reverse first Zagreb index, we have

$$\begin{array}{ccc}\hfill C{M}_1(PETIM)& =& \sum _{uvϵE(G)}(c_u+c_v)\hfill \\ & =& (1+2)(2^{n+1})+(2+2)(2^{n+4}-18)+(2+3)({6.2}^n-6)\hfill \\ & =& {4.2}^{n+4}+{3.2}^{n+1}+{3.2}^n-102.\hfill \end{array}$$

(ii)

From the definition of reverse second Zagreb index, we have

$$\begin{array}{ccc}\hfill C{M}_2(PETIM)& =& \sum _{uvϵE(G)}(c_u.c_v)\hfill \\ & =& (1.2)(2^{n+1})+(2.2)(2^{n+4}-18)+(2.3)({6.2}^n-6)\hfill \\ & =& {4.2}^{n+4}+{2.2}^{n+1}+{36.2}^n-108.\hfill \end{array}$$

 □

Theorem 6.

The first and second reverse Zagreb polynomials of $(PETIM)$ are,

1. 

$C{M}_1(PETIM,x)=({6.2}^n-6)x^5+(2^{(n+4)}-18)x^4+(2^{n+1})x^3.$

2. 

$C{M}_2(PETIM,x)=({6.2}^n-6)x^6+(2^{(n+4)}-18)x^4+(2^{n+1})x^2.$

Proof. 

(i)

From the information given in Theorem 5 and by the definition of reverse first Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill C{M}_1(PETIM,x)& =& \sum _{uvϵE(G)}{x}^{(c_u+c_v)}\hfill \\ & =& (2^{n+1})x^{(1+2)}+(2^{(n+4)}-18)x^{(2+2)}+({6.2}^n-6)x^{(2+3)}\hfill \\ & =& ({6.2}^n-6)x^5+(2^{(n+4)}-18)x^4+(2^{n+1})x^3.\hfill \end{array}$$

(ii)

From the information given in Theorem 5 and by the definition of reverse second Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill C{M}_2(PETIM,x)& =& \sum _{uvϵE(G)}{x}^{(c_u.c_v)}\hfill \\ & =& (2^{n+1})x^{(1.2)}+(2^{(n+4)}-18)x^{(2.2)}+({6.2}^n-6)x^{(2.3)}\hfill \\ & =& ({6.2}^n-6)x^6+(2^{(n+4)}-18)x^4+(2^{n+1})x^2.\hfill \end{array}$$

 □

Theorem 7.

The first and second reverse hyper-Zagreb indices of prophyrin Dendrimer $PETIM$ are

1. 

$HC{M}_1(PETIM)={16.2}^{n+4}+{9.2}^{n+1}+{150.2}^n-438,$

2. 

$HC{M}_2(PETIM)={16.2}^{n+4}+{4.2}^{n+1}+{216.2}^n-504.$

Proof. 

(i)

From the information given in Theorem 5 and by the definition of reverse first hyper-Zagreb index, we have

$$\begin{array}{ccc}\hfill HC{M}_1(PETIM)& =& \sum _{uvϵE(G)}{(c_u+c_v)}^2\hfill \\ & =& {(1+2)}^2(2^{n+1})+{(2+2)}^2(2^{n+4}-18)+{(2+3)}^2({6.2}^n-6)\hfill \\ & =& {16.2}^{n+4}+{9.2}^{n+1}+{150.2}^n-438.\hfill \end{array}$$

(ii)

From the information given in Theorem 5 and by the definition of reverse second hyper-Zagreb index, we have

$$\begin{array}{ccc}\hfill HC{M}_2(PETIM)& =& \sum _{uvϵE(G)}{(c_u.c_v)}^2\hfill \\ & =& {(1.2)}^2(2^{n+1})+{(2.2)}^2(2^{n+4}-18)+{(2.3)}^2({6.2}^n-6)\hfill \\ & =& {16.2}^{n+4}+{4.2}^{n+1}+{216.2}^n-504.\hfill \end{array}$$

 □

Theorem 8.

The first and second reverse hyper-Zagreb polynomials of $PETIM$ are

1. 

$HC{M}_1(PETIM,x)=({6.2}^n-6)x^{25}+(2^{(n+4)}-18)x^{16}+(2^{n+1})x^9.$

2. 

$HC{M}_2(PETIM,x)=({6.2}^n-6)x^{36}+(2^{(n+4)}-18)x^{16}+(2^{n+1})x^4.$

Proof. 

(i)

From the information given in Theorem 5 and by the definition of reverse first hyper-Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill HC{M}_1(PETIM,x)& =& \sum _{uvϵE(G)}{x}^{{(c_u+c_v)}^2}\hfill \\ & =& (2^{n+1})x^{{(1+2)}^2}+(2^{n+4}-18)x^{{(2+2)}^2}+({6.2}^n-6)x^{{(2+3)}^2}\hfill \\ & =& ({6.2}^n-6)x^{25}+(2^{(n+4)}-18)x^{16}+(2^{n+1})x^9.\hfill \end{array}$$

(ii)

From the information given in Theorem 5 and by the definition of reverse second hyper-Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill HC{M}_2(PETIM,x)& =& \sum _{uvϵE(G)}{x}^{{(c_u.c_v)}^2}\hfill \\ & =& (2^{n+1})x^{{(1.2)}^2}+(2^{n+4}-18)x^{{(2.2)}^2}+({6.2}^n-6)x^{{(2.3)}^2}\hfill \\ & =& ({6.2}^n-6)x^{36}+(2^{(n+4)}-18)x^{16}+(2^{n+1})x^4.\hfill \end{array}$$

 □

The values of first and second reverse Zagreb indices and first and second reverse hyper-Zagreb indices of $PETIM$ for specific values of n are given in

Table 2. Topological indices of $PETIM$.

	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
First Reverse Zagreb Index	44	190	482	1066	2234	4570	9242	18,586	37,274
Second Reverse Zagreb Index	100	308	724	1556	3220	6548	13,204	26,516	53,140
First Reverse Hyper-Zagreb Index	410	1258	2954	6346	13,130	26,698	53,834	108,106	216,650
Second Reverse Hyper-Zagreb Index	456	1416	3336	7176	14,856	30,216	60,936	122,376	245,256

.

3.3. Zinc Prophyrin Dendrimer $DP{Z}_n$

In this section, we compute reverse Zagreb indices, reverse Zagreb polynomials, reverse hyper-Zagreb indices and reverse hyper-Zagreb polynomials of Zinc Prophyrin Dendrimer $DP{Z}_n$.

Theorem 9.

Let $DP{Z}_n$ be a Zinc Prophyrin Dendrimer. Then, the first and second reverse Zagreb indices are

1. 

$C{M}_1(DP{Z}_n)={328.2}^n-156,$

2. 

$C{M}_2(DP{Z}_n)={416.2}^n-188.$

Proof. 

In Zinc Prophyrin dendrimer $DP{Z}_n$, there are $96n-10$ vertices and $105n-11$ edges. The edge set of $DP{Z}_n$ can be divided into following four classes by mean of the degree of end vertices.

$$E_1(DP{Z}_n)=\{uvϵE(DP{Z}_n);d_u=2,d_v=2\},$$

$$E_2(DP{Z}_n)=\{uvϵE(DP{Z}_n);d_u=2,d_v=3\},$$

$$E_3(DP{Z}_n)=\{uvϵE(DP{Z}_n);d_u=3,d_v=3\},$$

$$E_4(DP{Z}_n)=\{uvϵE(DP{Z}_n);d_u=3,d_v=4\}.$$

In Figure 3, one can count easily that $|E_1(DP{Z}_n)|={16.2}^n-4$, $|E_2(DP{Z}_n)|={40.2}^n-16$, $|E_3(DP{Z}_n)|={8.2}^n-16$ and $|E_4(DP{Z}_n)|=4.$

The maximum vertex degree $\Delta (G)$ of $DP{Z}_n$ is 4, so

$$C{E}_1(DP{Z}_n)=\{uvϵE(DP{Z}_n);d_u=3,d_v=3\},$$

$$C{E}_2(DP{Z}_n)=\{uvϵE(DP{Z}_n);d_u=3,d_v=2\},$$

$$C{E}_3(DP{Z}_n)=\{uvϵE(DP{Z}_n);d_u=2,d_v=2\},$$

$$C{E}_4(DP{Z}_n)=\{uvϵE(DP{Z}_n);d_u=2,d_v=1\}.$$

In addition, $|E_1(DP{Z}_n)|={16.2}^n-4$, $|E_2(DP{Z}_n)|={40.2}^n-16$, $|E_3(DP{Z}_n)|={8.2}^n-16$ and $|E_4(DP{Z}_n)|=4.$

(i)

Now, from the definition of reverse first Zagreb index, we have

$$\begin{array}{ccc}\hfill C{M}_1(DP{Z}_n)& =& \sum _{uvϵE(G)}(c_u+c_v)\hfill \\ & =& (3+3)({16.2}^n-4)+(3+2)({40.2}^n-16)\hfill \\ & & +(2+2)({8.2}^n-16)+(2+1)(4)\hfill \\ & =& {328.2}^n-156.\hfill \end{array}$$

(ii)

From the definition of reverse second Zagreb index, we have

$$\begin{array}{ccc}\hfill C{M}_2(DP{Z}_n)& =& \sum _{uvϵE(G)}(c_u.c_v)\hfill \\ & =& (3.3)({16.2}^n-4)+(3.2)({40.2}^n-16)\hfill \\ & & +(2.2)({8.2}^n-16)+(2.1)(4)\hfill \\ & =& {416.2}^n-188.\hfill \end{array}$$

 □

Theorem 10.

The first and second reverse Zagreb polynomials of $(DP{Z}_n)$ are

1. 

$C{M}_1(DP{Z}_n,x)=({16.2}^n-4)x^6+({40.2}^n-16)x^5+({8.2}^n-16)x^4+(4)x^3,$

2. 

$C{M}_2(DP{Z}_n,x)=(16.2^n-4)x^9+(40.2^n-16)x^6+({8.2}^n-16)x^4+(4)x^2.$

Proof. 

(i)

From the information given in Theorem 9 and by the definition of reverse first Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill C{M}_1(DP{Z}_n,x)& =& \sum _{uvϵE(G)}{x}^{(c_u+c_v)}\hfill \\ & =& ({16.2}^n-4)x^{(3+3)}+({40.2}^n-16)x^{(3+2)}\hfill \\ & & +({8.2}^n-16)x^{(2+2)}+(4)x^{(2+1)}\hfill \\ & =& ({16.2}^n-4)x^6+({40.2}^n-16)x^5+({8.2}^n-16)x^4+(4)x^3.\hfill \end{array}$$

(ii)

From the information given in Theorem 9 and by the definition of reverse second Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill C{M}_2(DP{Z}_n,x)& =& \sum _{uvϵE(G)}{x}^{(c_u.c_v)}\hfill \\ & =& ({16.2}^n-4)x^{(3.3)}+({40.2}^n-16)x^{(3.2)}\hfill \\ & & +({8.2}^n-16)x^{(2.2)}+(4)x^{(2.1)}\hfill \\ & =& ({16.2}^n-4)x^9+({40.2}^n-16)x^6+({8.2}^n-16)x^4+(4)x^2.\hfill \end{array}$$

 □

Theorem 11.

Let $DP{Z}_n$ be a Zinc Prophyrin Dendrimer, the first and second reverse hyper-Zagreb indices are

1. 

$HC{M}_1(DP{Z}_n)={1704.2}^n-764,$

2. 

$HC{M}_2(DP{Z}_n)={2964.2}^n-1140.$

Proof. 

(i)

From the information given in Theorem 9 and by the definition of reverse first hyper-Zagreb index, we have

$$\begin{array}{ccc}\hfill HC{M}_1(DP{Z}_n)& =& \sum _{uvϵE(G)}{(c_u+c_v)}^2\hfill \\ & =& {(3+3)}^2({16.2}^n-4)+{(3+2)}^2({40.2}^n-16)\hfill \\ & & +{(2+2)}^2({8.2}^n-16)+{(2+1)}^2(4)\hfill \\ & =& {1704.2}^n-764.\hfill \end{array}$$

(ii)

From the information given in Theorem 9 and by the definition of reverse first hyper-Zagreb index, we have

$$\begin{array}{ccc}\hfill HC{M}_2(DP{Z}_n)& =& \sum _{uvϵE(G)}{(c_u.c_v)}^2\hfill \\ & =& {(3.3)}^2({16.2}^n-4)+{(3.2)}^2({40.2}^n-16)\hfill \\ & & +{(2.2)}^2({8.2}^n-16)+{(2.1)}^2(4)\hfill \\ & =& {2964.2}^n-1140.\hfill \end{array}$$

 □

Theorem 12.

The first and second reverse hyper-Zagreb polynomials of $(DP{Z}_n)$ are

1. 

$HC{M}_1(DP{Z}_n,x)=({16.2}^n-4)x^{36}+({40.2}^n-16)x^{25}+({8.2}^n-16)x^{16}+(4)x^9,$

2. 

$HC{M}_2(DP{Z}_n,x)=({16.2}^n-4)x^{81}+({40.2}^n-16)x^{36}+({8.2}^n-16)x^{16}+(4)x^4.$

Proof. 

(i)

From the information given in Theorem 9 and by the definition of reverse first hyper-Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill HC{M}_1(DP{Z}_n,x)& =& \sum _{uvϵE(G)}{x}^{{(c_u+c_v)}^2}\hfill \\ & =& ({16.2}^n-4)x^{{(3+3)}^2}+({40.2}^n-16)x^{{(3+2)}^2}\hfill \\ & & +({8.2}^n-16)x^{{(2+2)}^2}+(4)x^{{(2+1)}^2}\hfill \\ & =& ({16.2}^n-4)x^{36}+({40.2}^n-16)x^{25}+({8.2}^n-16)x^{16}+(4)x^9.\hfill \end{array}$$

(ii)

From the information given in Theorem 9 and by the definition of reverse second hyper-Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill HC{M}_2(DP{Z}_n,x)& =& \sum _{uvϵE(G)}{x}^{{(c_u.c_v)}^2}\hfill \\ & =& ({16.2}^n-4)x^{{(3.3)}^2}+({40.2}^n-16)x^{{(3.2)}^2}\hfill \\ & & +({8.2}^n-16)x^{{(2.2)}^2}+(4)x^{{(2.1)}^2}\hfill \\ & =& ({16.2}^n-4)x^{81}+({40.2}^n-16)x^{36}+({8.2}^n-16)x^{16}+(4)x^4.\hfill \end{array}$$

 □

The values of first and second reverse Zagreb indices and first and second reverse hyper-Zagreb indices of $(DP{Z}_n)$ for specific values of n are given in

Table 3. Topological indices of $(DP{Z}_n)$.

	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
First Reverse Zagreb Index	500	1156	2468	5092	10,340	20,836	41,828	83,812	167,780
Second Reverse Zagreb Index	644	1476	3140	6468	13,124	26,436	53,060	106,308	212,804
First Reverse Hyper-Zagreb Index	2644	6052	12,868	26,500	53,764	108,292	217,348	435,460	871,684
Second Reverse Hyper-Zagreb Index	4788	10,716	22,572	46,284	93,708	188,556	378,252	757,644	1,516,428

.

3.4. Poly(EThylene Amide Amine) Dendrimer $PETAA$

In this section, we compute reverse Zagreb indices, reverse Zagreb polynomials, reverse hyper-Zagreb indices and reverse hyper-Zagreb polynomials of Poly(EThylene Amide Amine) Dendrimer $PETAA$.

Theorem 13.

Let $PETAA$ be a Poly(EThylene Amide Amine) Dendrimer, then the first and second reverse Zagreb indices are

1. 

$C{M}_1(PETAA)={100.2}^n-67.$

2. 

$C{M}_2(PETAA)={100.2}^n-56.$

Proof. 

In Poly(EThylene Amide Amine) dendrimer $PETAA$, there are ${44.2}^n-18$ vertices and ${44.2}^n-19$ edges. Based on the degree of end vertices, the edge set of $PETAA$ can be divided into following four classes.

$$E_1(PETAA)=\{uvϵE(PETAA);d_u=1,d_v=2\},$$

$$E_2(PETAA)=\{uvϵE(PETAA);d_u=1,d_v=3\},$$

$$E_3(PETAA)=\{uvϵE(PETAA);d_u=2,d_v=2\},$$

$$E_4(PETAA)=\{uvϵE(PETAA);d_u=2,d_v=3\}.$$

In Figure 4, one can count easily that $|E_1(PETAA)|={4.2}^n$, $|E_2(PETAA)|={4.2}^n-2$, $|E_3(PETAA)|={16.2}^n-8$ and $|E_4(PETAA)|={20.2}^n-9$.

The maximum vertex degree $\Delta (G)$ of $PETAA$ is 3. Thus,

$$C{E}_1(PETAA)=\{uvϵE(PETAA);d_u=3,d_v=2\},$$

$$C{E}_2(PETAA)=\{uvϵE(PETAA);d_u=3,d_v=1\},$$

$$C{E}_3(PETAA)=\{uvϵE(PETAA);d_u=2,d_v=2\},$$

$$C{E}_4(PETAA)=\{uvϵE(PETAA);d_u=2,d_v=1\}.$$

In addition, $|C{E}_1(PETAA)|={4.2}^n$, $|C{E}_2(PETAA)|={4.2}^n-2$, $|C{E}_3(PETAA)|={16.2}^n-8$ and $|C{E}_4(PETAA)|={20.2}^n-9$.

(i)

Now, from the definition of reverse first Zagreb index, we have

$$\begin{array}{ccc}\hfill C{M}_1(PETAA)& =& \sum _{uvϵE(G)}(c_u+c_v)\hfill \\ & =& (3+2)({4.2}^n)+(3+1)({4.2}^n-2)+\hfill \\ & & (2+2)({16.2}^n-8)+(2+1)({20.2}^n-9)\hfill \\ & =& {100.2}^n-67.\hfill \end{array}$$

(ii)

From the definition of reverse Zagreb index, we have

$$\begin{array}{ccc}\hfill C{M}_2(PETAA)& =& \sum _{uvϵE(G)}(c_u.c_v)\hfill \\ & =& (3.2)({4.2}^n)+(3.1)({4.2}^n-2)+\hfill \\ & & (2.2)({16.2}^n-8)+(2.1)({20.2}^n-9)\hfill \\ & =& {100.2}^n-56.\hfill \end{array}$$

Theorem 14.

The first and second reverse Zagreb polynomal of Poly(EThylene Amide Amine) dendrimer $PETAAA$ are

1. 

$C{M}_1(PETAA,x)=({4.2}^n)x^5+({20.2}^n-10)x^4+({20.2}^n-9)x^3,$

2. 

$C{M}_2(PETAA,x)=({4.2}^n)x^6+({16.2}^n-8)x^4+({24.2}^n-11)x^3.$

Proof. 

(i)

From the information given in Theorem 13 and by the definition of reverse first Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill C{M}_1(PETAA,x)& =& \sum _{uvϵE(G)}{x}^{(c_u+c_v)}\hfill \\ & =& ({4.2}^n)x^{(3+2)}+({4.2}^n-2)x^{(3+1)}+\hfill \\ & & ({16.2}^n-8)x^{(2+2)}+({20.2}^n-9)x^{(2+1)}\hfill \\ & =& ({4.2}^n)x^5+({20.2}^n-10)x^4+({20.2}^n-9)x^3.\hfill \end{array}$$

(ii)

From the information given in Theorem 13 and by the definition of reverse second Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill C{M}_2(PETAA,x)& =& \sum _{uvϵE(G)}{x}^{(c_u.c_v)}\hfill \\ & =& ({4.2}^n)x^{(3.2)}+({4.2}^n-2)x^{(3.1)}+\hfill \\ & & ({16.2}^n-8)x^{(2.2)}+({20.2}^n-9)x^{(2.1)}\hfill \\ & =& ({4.2}^n)x^6+({16.2}^n-8)x^4+({24.2}^n-11)x^3.\hfill \end{array}$$

 □

Theorem 15.

Let $PETAA$ be a Poly (EThylene Amide Amine) Dendrimer. Then, the first and second reverse hyper-Zagreb indices are

1. 

$HC{M}_1(PETAA)={420.2}^n-241,$

2. 

$HC{M}_2(PETAA)={436.2}^n-182.$

Proof. 

(i)

From the information given in Theorem 13 and by the definition of reverse first hyper-Zagreb index, we have

$$\begin{array}{ccc}\hfill HC{M}_1(PETAA)& =& \sum _{uvϵE(G)}{(c_u+c_v)}^2\hfill \\ & =& {(3+2)}^2({4.2}^n)+{(3+1)}^2({4.2}^n-2)+\hfill \\ & & {(2+2)}^2({16.2}^n-8)+{(2+1)}^2({20.2}^n-9)\hfill \\ & =& {420.2}^n-241.\hfill \end{array}$$

(ii)

From the information given in Theorem 13 and by the definition of reverse second hyper-Zagreb index, we have

$$\begin{array}{ccc}\hfill HC{M}_2(PETAA)& =& \sum _{uvϵE(G)}{(c_u.c_v)}^2\hfill \\ & =& {(3.2)}^2({4.2}^n)+{(3.1)}^2({4.2}^n-2)+\hfill \\ & & {(2.2)}^2({16.2}^n-8)+{(2.1)}^2({20.2}^n-9)\hfill \\ & =& {436.2}^n-182.\hfill \end{array}$$

 □

Theorem 16.

The first and second reverse hyper-Zagreb polynomal of Poly(EThylene Amide Amine) dendrimer $PETAAA$ are

1. 

$HC{M}_1(PETAA,x)=({4.2}^n)x^{25}+({20.2}^n-10)x^{16}+({20.2}^n-9)x^9,$

2. 

$HC{M}_2(PETAA,x)=({4.2}^n)x^{36}+({16.2}^n-8)x^{16}+({24.2}^n-11)x^9.$

Proof. 

(i)

From the information given in Theorem 13 and by the definition of reverse first hyper-Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill HC{M}_1(PETAA,x)& =& \sum _{uvϵE(G)}{x}^{{(c_u+c_v)}^2}\hfill \\ & =& (4.2^n)x^{{(3+2)}^2}+({4.2}^n-2)x^{{(3+1)}^2}+\hfill \\ & & ({16.2}^n-8)x^{{(2+2)}^2}+({20.2}^n-9)x^{{(2+1)}^2}\hfill \\ & =& ({4.2}^n)x^{25}+({20.2}^n-10)x^{16}+({20.2}^n-9)x^9.\hfill \end{array}$$

(ii)

From the information given in Theorem 13 and by the definition of reverse second hyper-Zagreb polynomial, we have

$$\begin{array}{ccc}\hfill HC{M}_2(PETAA,x)& =& \sum _{uvϵE(G)}{x}^{{(c_u.c_v)}^2}\hfill \\ & =& ({4.2}^n)x^{{(3.2)}^2}+({4.2}^n-2)x^{{(3.1)}^2}+\hfill \\ & & ({16.2}^n-8)x^{{(2.2)}^2}+({20.2}^n-9)x^{{(2.1)}^2}\hfill \\ & =& ({4.2}^n)x^{36}+({16.2}^n-8)x^{16}+({24.2}^n-11)x^9.\hfill \end{array}$$

 □

The values of first and second reverse Zagreb indices and first and second reverse hyper-Zagreb indices of Poly(EThylene Amide Amine) dendrimer for specific values of n are given in

Table 4. Topological indices of Poly (EThylene Amide Amine) dendrimer.

	n = 1	n = 2	n = 3	n = 4	n = 5	n = 6	n = 7	n = 8	n = 9
First Reverse Zagreb Index	133	333	733	1533	3133	6333	12,733	25,533	51,133
Second Reverse Zagreb Index	144	344	744	1544	3144	6344	12,744	25,544	51,144
First Reverse Hyper-Zagreb Index	599	1439	3119	6479	13,199	26,639	53,519	107,279	214,799
Second Reverse Hyper-Zagreb Index	690	1562	3306	6794	13,770	27,722	55,626	111,434	223,050

.

4. Graphical Comparison and Concluding Remarks

There are many application of dendrimers, typically involve conjugating other chemical species to the dendrimer surface that can function as detecting agents (such as a dye molecule), affinity ligands, targeting components, radioligands, imaging agents, or pharmaceutically active compounds. Topological indices of dendrimers are useful in theoretical chemistry, pharmacology, toxicology, and environmental chemistry [56,57]. In this paper, we compute reverse first Zagreb index, reverse second Zagreb index, reverse first hyper-Zagreb index, reverse second hyper-Zagreb index, reverse first Zagreb polynomial, reverse second Zargeb polynomial, reverse first hyper-Zagreb polynomial and reverse second hyper-Zagreb polynomial of Prophyrin, Propyl ether imine, Zinc Porphyrin and Poly(ethylene amido amine) dendrimers. Figure 5 shows that Zinc Porphyrin dendrimers get highest value of first reverse Zagreb index and Prophyrin get least value of first reverse Zagreb index. In Figure 6, Figure 7 and Figure 8, we can choose the dendrimers having largest and least values of second reverse Zagreb, first reverse hyper-Zagreb and second reverse hyper-Zagreb index, respectively.