The Theta Polynomial Θ(G,x) and the Theta Index Θ(G) of Molecular Graph Polycyclic Aromatic Hydrocarbons PAHk

3934 | P a g e N o v a m b e r 1 1 , 2 0 1 5 The Theta Polynomial Θ(G,x) and the Theta Index Θ(G) of Molecular Graph Polycyclic Aromatic Hydrocarbons PAHk Wei Gao , Mohammad Reza Farahani 1.School of Information Science and Technology, Yunnan Normal University, Kunming 650500, China. 2. Department of Applied Mathematics of Iran University of Science and Technology (IUST), Narmak, Tehran 16844, Iran.


INTRODUCTION
Let G be a simple molecular graph without directed and multiple edges and without loops, the vertex and edge-sets of which are represented by V(G) and E(G), respectively. A graph can be described by a connection table, a sequence of numbers, a matrix, a polynomial or by a single number (often called a topological index). A counting polynomial can be written as: with the exponents showing the extent of partitions p(G), p(G)=P(G) of a graph property P(G) while the coefficients m(G, k) are related to the number of partitions of extent k.
Let G(V,E) be a connected graph, two edges e=uv and f=xy of G are called co-distant: e co f, if and only if d(u,x)=d(v,y)=k and d(u,y)=d(v,x)=k+1 or vice versa, for a non-negative integer k.
If co is an equivalence relation: [1][2][3] Then, C(e):={f∈E(G)| f co e} is the set of edges in G, co-distant to the edge e∈E(G) and G is called a co-graph. Consequently, C(e) is called an orthogonal cut set ocs of G and E(G) is the union of disjoint orthogonal cuts: for i≠j and i, j=1, 2, …, k. The relation ops is not necessarily transitive. Observe an ops is an ocs only in partial cubes.
Observe co is a θ relation, (Djokovic-Winkler relation) [4,5] and G is a co-graph if and only if it is a partial cube, a result due to Klavžar [6]. In a plane bipartite graph, an edge e is in relation θ with any opposite edge f if the faces of the plane graph are isometric (which is the case of the most chemical graphs). Then an orthogonal cut oc with respect to a given edge is the smallest subset of edges closed under this operation and C(e) is precisely a θ-class of G. If ops is an ocs, as in partial cubes, we can write the following counting polynomials [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]: x)and Θ(G,x) count equidistant edges in G while Sd(G, X ) and Π(G,x), count non-equidistant edges. The first two polynomials are counted once for a strip while the last two are counted for each edge, so that the coefficients are multiplied with k.
In this present study, we compute the Theta polynomial Θ(G,x) and the Theta index Θ(G) of molecular graph Polycyclic Aromatic Hydrocarbons PAHk, for all positive integer number k.

MAIN RESULTS AND DISCUSSION
The Polycyclic Aromatic Hydrocarbons PAHk for all positive integer number k is ubiquitous combustion products. They have been implicated as carcinogens and play a role in graphitization of organic materials [24]. In addition, they are of interest as molecular analogues of graphite [25] as candidates for interstellar species [26] and as building blocks of functional materials for device applications [25][26][27]. Synthetic routes to Polycyclic Aromatic Hydrocarbons PAHk are available [28] and a detailed knowledge of all these features would therefore be necessary for the tuning of molecular properties towards specific applications. N o v a m b e r 11, 2 0 1 5 Reader can see some first members of this family in Figure 1. In references [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41] some properties and more historical details of this family of hydrocarbon molecules are studded. Now, we counting all opposite edge strips ops m(PAHk,c) of the general representation of Polycyclic Aromatic Hydrocarbons PAHk, by using the Cut Method. The Cut Method and its general form studied by S. Klavzar [42].
By using the Cut Method, we see that the Polycyclic Aromatic Hydrocarbons is a co-graph and from Figure 2, one can see that there are k+1 distinct case of qoc strips for PAHk such that the size of a qoc strip Ci for i=1,..,k-1 is equal to k+i (=|Ci|=ci) and for i=0, |C0|=k.
In other words, , , Here the proof of theorem is completed.■

ACKNOWLEDGMENTS
We thank the reviewers for their constructive comments in improving the quality of this paper. This work was supported in part by NSFC (no. 11401519).