Some Reverse Degree-Based Topological Indices and Polynomials of Dendrimers

: 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


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].
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.

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 ∆(G). Kulli [20] introduced the concept of reverse vertex degree c v , as c v = ∆(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 |X| 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 Zagreb indices help us in finding Π 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:

respectively.
Shirdel et al. [55] proposed the first and second hyper-Zagreb indices as: and Motivated by these indices, the first and second reverse Zagreb indices was defined in [20] as: The first and second Reverse hyper-Zagreb indices was also defined in the same paper as: and 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: and the first and second hyper-Zagreb polynomials are defined as: and

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.
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 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 ∆(G) in D n P n is 4, so we have following six types of reverse edges in D n P n .
(i) Now, using the definition of reverse first Zagreb index, we have (ii) Using the definition of reverse second Zagreb index, we have Theorem 2. The first and second reverse Zagreb polynomials of D n P n are Proof.
(i) Using the information given in Theorem 1 and definition of reverse first Zagreb polynomial, we have (ii) Using the information given in Theorem 1 and definition of reverse second Zagreb polynomial, we have Theorem 3. The first and second reverse hyper-Zagreb indices of prophyrin Dendrimer D n P n are 1. Proof.
(i) Using the information given in Theorem 1 and definition of reverse first hyper-Zagreb index, we have (ii) Using the information given in Theorem 1 and definition of reverse second hyper-Zagreb index, we have Theorem 4. The first and second reverse hyper-Zagreb polynomials of D n P n are 1. Proof.
(i) Using the information given in Theorem 1 and definition of reverse first hyper-Zagreb polynomial, we have (ii) Using the information given in Theorem 1 and definition of reverse second hyper-Zagreb polynomial, we have 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.
Proof. In Propyl Ether Imine dendrimer PETI M, there are 24.2 n − 23 vertices and 24.2 n − 24 edges. Based on the degree of end vertices, the edge set of PETI M can be divided into following three classes.
The maximum vertex degree ∆(G) of PETI M is 3, so we have following types of reverse edges.
(i) Now, from the definition of reverse first Zagreb index, we have (ii) From the definition of reverse second Zagreb index, we have Theorem 6. The first and second reverse Zagreb polynomials of (PETI M) are, Proof.
(i) From the information given in Theorem 5 and by the definition of reverse first Zagreb polynomial, we have (ii) From the information given in Theorem 5 and by the definition of reverse second Zagreb polynomial, we have Proof.
(i) From the information given in Theorem 5 and by the definition of reverse first hyper-Zagreb index, we have (ii) From the information given in Theorem 5 and by the definition of reverse second hyper-Zagreb index, we have Proof.
(i) From the information given in Theorem 5 and by the definition of reverse first hyper-Zagreb polynomial, we have (ii) From the information given in Theorem 5 and by the definition of reverse second hyper-Zagreb polynomial, we have The values of first and second reverse Zagreb indices and first and second reverse hyper-Zagreb indices of PETI M for specific values of n are given in Table 2. Proof. In Zinc Prophyrin dendrimer DPZ n , there are 96n − 10 vertices and 105n − 11 edges. The edge set of DPZ n can be divided into following four classes by mean of the degree of end vertices.
In Figure 3, one can count easily that The maximum vertex degree ∆(G) of DPZ n is 4, so In addition, |E 1 (DPZ n )| = 16. Proof.
(i) From the information given in Theorem 9 and by the definition of reverse first Zagreb polynomial, we have (ii) From the information given in Theorem 9 and by the definition of reverse second Zagreb polynomial, we have (ii) From the information given in Theorem 9 and by the definition of reverse first hyper-Zagreb index, we have Theorem 12. The first and second reverse hyper-Zagreb polynomials of (DPZ n ) are

Proof.
(i) From the information given in Theorem 9 and by the definition of reverse first hyper-Zagreb polynomial, we have (ii) From the information given in Theorem 9 and by the definition of reverse second hyper-Zagreb polynomial, we have The values of first and second reverse Zagreb indices and first and second reverse hyper-Zagreb indices of (DPZ n ) for specific values of n are given in Table 3. Table 3. Topological indices of (DPZ n ).
The maximum vertex degree ∆(G) of PETAA is 3. Thus,

Proof.
(i) From the information given in Theorem 13 and by the definition of reverse first Zagreb polynomial, we have x (c u +c v ) = (4. Proof. (i) From the information given in Theorem 13 and by the definition of reverse first hyper-Zagreb index, we have = 420.2 n − 241.
(ii) From the information given in Theorem 13 and by the definition of reverse second hyper-Zagreb index, we have Proof.
(i) From the information given in Theorem 13 and by the definition of reverse first hyper-Zagreb polynomial, we have (ii) From the information given in Theorem 13 and by the definition of reverse second hyper-Zagreb polynomial, we have 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.

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 Figures 6-8, 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.