Schultz and Modified Schultz Polynomials of Coronene Polycyclic Aromatic Hydrocarbons

Let G = ( V;E ) be a simple connected graph. The sets of vertices and edges of G are denoted by V = V ( G ) and E = E ( G ), respectively. In such a simple molecular graph, vertices represent atoms and edges represent bonds. The distance between the vertices u and v in V ( G ) of graph G is the number of edges in a shortest path connecting them, we denote by d ( u,v ). In graph theory, we have many invariant polynomials for a graph G . In this research, we computing the Schultz polynomial, Modified Schultz polynomial, Hosoya polynomial and their topological indices of a Hydrocarbon molecule, that we call “Coronene Polycyclic Aromatic Hydrocarbons”.


INTRODUCTION
Let G be a connected graph. The vertex-set and edge-set of G denoted by V(G) and E(G) respectively, such that in the connected molecular graph G, vertices represent atoms and edges represent bonds. In graph theory, the degree of a vertex vV(G) is the number of vertices joining to v and denoted by d v (G) (or simpely d v ). If e is an edge of G, connecting the vertices u and v, then we write e = uv. Also d(u,v) = d (u,v|G) is the distance between vertices u and v is equal to the length of the shortest path that connects them in G.
In chemical graph theory, we have invariant polynomials and topological indices for any molecular graphs. Such that topological indices of molecular graphs and nanostructures are numerical descriptors that are derived from graph of chemical compounds. Such indices based on the distances in graph are widely used for establishing relationships between the structure of molecular graphs and their physicochemical properties.
Usage of topological indices in Biology and Chemistry began in 1947 when chemist Harold Wiener [1] introduced Wiener index to demonstrate correlations between physicochemical properties of organic compounds of molecular graphs. Wiener originally defined his index (W) on trees and studied its use for correlations of physico-chemical properties of alkanes, alcohols, amines and their analogous compounds [2]. The Wiener index of a graph G is denoted by W(G) and defined as the sum of distances between all pairs of vertices in simple graph G: Another based structure descriptors is Schultz index, the Schultz index (Sc) was introduced by Harry P. Schultz in 1989 [4-6], as the "molecular topological index" and it is defined by: where d u and d v are degrees of vertices u and v.
On based the Schultz index S. Klavžar and I. Gutman introduced the Modified Schultz index in 1997 [7][8][9] and defined as: In chemical graph theory, there are two important polynomials for these structure descriptors and "Schultz polynomial" and "Modified Schultz polynomial" of G are defined respectively as: These based structure descriptors and their polynomials studied and computed in many papers  and also the Hosoya polynomial and Wiener index of some molecular graph computed [7,12,13,23,[36][37][38][39][40][41][42][43]. In this research, we computing the Schultz polynomial, Modified Schultz polynomial, Hosoya polynomial and their topological indices of a Hydrocarbon molecule, that we call "Coronene Polycyclic Aromatic Hydrocarbons".

RESULTS AND DISCUSSION
In chemical, physics and nano sciences, we have the appealing structure, especially symmetric structure with chemical constitution purporting. One of the molecular graphs is Benzenoid System. Since molecule benzene (C 6 H 6 ) is more practical in the chemical, physics and nano science, we lionize its structure.

CONCLUSION
In this paper, counting polynomials called "Schultz