Zagreb Polynomials and redefined Zagreb indices of Dendrimers and Polyomino Chains

Abstract Dendrimers have an incredibly strong potential because their structure allows multivalent frameworks, i.e. one dendrimer molecule has many possible destinations to couple to a functioning species. Researchers expected to utilize the hydrophobic conditions of the dendritic media to lead photochemical responses that make the things that are artificially tested. Carboxylic acid and phenol- terminated water-dissolvable dendrimers were joined to set up their utility in tranquilize conveyance and furthermore driving compound reactions in their inner parts. This may empower scientists to associate both concentrating on atoms and medication particles to the equivalent dendrimer, which could diminish negative manifestations of prescriptions on sound and health cells. Topological indices are numerical numbers associated with the graphs of dendrimers and are invariant up to graph isomorphism. These numbers compare certain physicochemical properties like boiling point, strain energy, stability, etc. of a synthetic compound. There are three main types of topological indices, i.e degree-based, distance-based and spectrum-based. In this paper, our aim is to compute some degree-based indices and polynomials for some dendrimers and polyomino chains. We computed redefined first, second and third Zagreb indices of PAMAM dendrimers PD1, PD2, and DS1 and linear Polyomino chain Ln , Zigzag Polyomino chain Zn, polyomino chain with n squares and of m segments Bn1 $B_{n}^{1}$and Bn2 $B_{n}^{2}$We also computed some Zagreb polynomials of understudy dendrimers and chains.


Introduction
In medicine mathematical modelling is used to understand the structure of new drugs, usually as an undirected graph where each vertex exhibits a molecule and each edge addresses a bond between atoms. A huge number of new drugs are made each year which then requires significant work to choose the pharmacological, compound and organic qualities of these new drugs. This is challenging for countries, in for example South America, Southeast Asia, Africa and India where the cost for gauging the biochemical properties is prohibitive.
It has been proven in numerous studies that there is a strong link between the properties of compounds and drugs with their molecular structure. The topological index (TI) defined on the structure of these compounds can help researchers to develop an understanding of the physical characteristics, chemical reactivity and biological activity [1,2]. Therefore, the study of TIs of chemical structures of drugs can provide a theoretical basis for the preparation of new drugs [3].
The oldest degree-based TIs were defined by Gutman in [14] and are known by different names such as Sag. Loeb group parameters, Zagreb group index. Nowadays these indices are known as first Zagreb index and second Zagreb index. Zagreb indices are used to studying chirality, heterogeneity, ZE isomers and molecular complexity and have potential relevance with multiple linear regression models. For detailed survey we refer to [14][15][16][17].
The Polyomino Chains is a finite 2-connected floor plan, where each inner face (or a unit) is encompassed by a square of length one. It is a union of cells connected by edges in a planar square lattice [18][19][20]: Dendrimer originates from the Greek word meaning "trees" [21,22] and are redundantly spread molecules. Dendrimers are commonly symmetrical about the center and generally display a circular three-dimensional morphology. The first dendrimer was made by Fritz Vögtle in [23] utilizing distinctive engineered techniques RG Denkewalter in Allied in [24,25] Donald Tomalia in Dow Chemical in [26] and [27,28] and in [29] by George R. Newkome 1990, Craig Hawker and Jean Fréchet presented a combination union strategy. The prevalence of dendrimers has significantly expanded and y 2005 there were in excess of 5,000 logical papers and patents. We aim to study some Zagreb polynomials and redefined Zagreb indices of Polyomino Chains and Dendrimers in this paper.

Basic Notions
In this section, we will give some definitions and basic theory of chemical graph theory.
Throughout this paper G means a connected simple graph, V(G) and E(G) represent the vertex set and edge set of G respectively. The degree of a vertex v ∈ V(G) is the number of vertices attached to it. The formulae for the first and second Zagreb indices are (cf. [14]) Bindusree et al. [33] defined the following Zagreb type polynomials.
The first, second and third redefined Zagreb indices were defined by Ranjini et al. in [34]. These indicators appear as ReZG .

Zagreb Polynomials and Redefined Zagreb indices of PAMAM Dendrimers
Polyamidoamine (PAMAM) dendrimers are hyperbranched polymers with unparalleled sub-atomic consistency, subatomic weight distribution, characterized size and shape qualities and a multifunctional terminal surface. These nanoscale polymers comprise an ethylenediamine center, a redundant fanning amidoamine inward structure and an essential amine terminal surface. Dendrimers are "grown" off a central core in an iterative assembling process, with each resulting venture speaking to another "generation" of dendrimer. Expanding generation (atomic weight) produce bigger sub-atomic measurements, double the quantity of responsive surface destinations, and around twofold the sub-atomic load of the first era. PAMAM dendrimers likewise expect a spheroidal, globular shape at Generation 4 or above. Their usefulness is promptly custom fitted, and their consistency, measure and profoundly responsive "sub-atomic Velcro" surfaces is keys to their utilization. Here we study 1 PD which is PAMAM dendrimers with trifunctional center unit created by n G dendrimer with n growth stages and the PAMAM dendrimers 2 PD with various centers produced by dendrimer generators with n growth stages. 1 DS is PAMAM dendrimers with n growth stages. The M-polynomials, first and second Zagreb indices, modified Zagreb index, generalized Randic index, inverse Randic index, symmetric division index, harmonic index, inverse sum index and augmented Zagreb index for some dendrimers and Polyomino chains were computed in [35].
In this paper we aim to compute Zagreb polynomials and redefined Zagreb indices of the same structures that were previously studied in [35].      x The edge set of the molecular graph of 1 PD PAMAM dendrimers has the following four classes depending on the degrees of end vertices.    n a n a ( ) ( )   n a n a

Zagreb Polynomials and Redefined Zagreb indices of Polyomino Chains
The Polyomino system is a finite graph which is 2-connected plane in which each inner cell is encircled by a square. In simple words, the Polyomino system is an edge-connected union of cells within the planar square lattice. Polyomino chain is an example of Polyomino system [35].
The remaining proof is similar to Theorem 1.   ( )