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Introduction
In discrete mathematics, graph theory in general, not only the study of different properties of objects but it also tells us about objects having same properties as investigating object. These properties of different objects if of main interest. In particular, graph polynomials related to graph are rich in information about subgraph of investigating graph. Mathematical tools like polynomials and topological based numbers play an important rule, these tools gives important information about properties of chemical compounds. We can find out many hidden information about compounds through theses tools. Multifold graph polynomials are present in literature, various of them turned out to be applicable in mathematical chemistry. Actually, topological index is a numeric quantity that tells us about the whole structure of graph. Commonly used topological indices are degree-based and distance-based topological indices. Theses tools helps us to study physical, chemical reactivities and biological properties [1, 2, 3, 4, 5]. In 1947, Winner, firstly introduce the concept of topological index while working on boiling point. In particular, Hosoya polynomial [6] palys an important in the area if distance-based topological indices, we can find out Winer index, Hyper winer index and Tratch-stankevich-zefirove index by Hosoya polynomial. Another important degree-based polynomial is M-polynomial [7] that was introduced by Emeric Deutsch and Sandi Klavaza in 2015. Mainly, M-polynomial helps us to gain many information related to degree based topological indices. [8, 9, 10, 11].
Definition 1. If G=(V,E) is a graph and vɛV, the dv(G) denoted the degree of v. Let mij(G), i, j ≥ 1, be the number of edges uv of G such that {dv(G),du(G)} = {i, j}. The M-polynomial [7] is defined as
$$M(G;x,y)=\sum_{i\leq j} m_{ij}(G)x^{i} y^{j}$$
The firstly introduced topological index was Winer index [12], some one named this winer index path number. While studying chemical graph structures, commonly studied index in winer index because of having huge number of application in chemical graph theory [13], [14]. Another oldest topological index is Randic index introduced by Milan Randic in 1975. The Randic index [15] is defined as:
After some time Ballobas and Erdos [16] in 1998,and Amic et al. [17] gives the concept of generalized Randic index. Mathematicians and chemists gain many results from this concept [18]. Many results related to generalized randic index have been discussed in [19].
Ghorbani and Azimi [21] in 2012, introduced two new variants of zagreb indices
Definition 2. The first multiple Zagreb index of a graph G can be defined as:
Here, we will compute different polynomials of flower graph f(n×m), namely, M-Polynomial, Forgotten Polynomials and with the help of these polynomials, we will discuss different variants of Flower graph f(n×m).
Theorem 1
Let f(n×m)be Flower graph, where n = {0,1,2,⋯} and m = {0,1,2,⋯}. Then the M-polynomial is
Proof. For Flower graph f(n×m). The vertex set and edge sets are respectively, |V(f(n×m))| = n(m-1) and |E(f(n×m))| = nm The possible vertex degrees are 2 and 4, the contributed edges are (2,2) (2,4) and (4,4). From above condition, we can divided edge set of f(n×m) into three partitions,
Proposition 2. Let f(n×m)be Flower graph, where n = {0,1,2, ⋯} and m = {0,1,2,⋯}. Then the Forgotten Polynomial and Forgotten Index / F-index is given as, respectively,