Omega and Cluj-Ilmenau Indices of Hydrocarbon Molecules “ Polycyclic Aromatic Hydrocarbons PAHk ”

A topological index is a numerical value associated with chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. In this paper, we computed the Omega and Cluj-Ilumenau indices of a very famous hydrocarbon named as Polycyclic Aromatic Hydrocarbons k PAH for all integer number k.


Introduction
If the edges e and f are codistant we write it as e co f.Relation co is reflexive and symmetric but generally not transitive.If co relation is transitive then it is an equivalence relation.A graph G in which co is an equivalence relation is called co-graph, and the subset of edges the edge set ( ) E G can be written as the union of disjoint orthogonal cuts, i.e.
( ) 1 2 ; for be two edges of G which are opposite or topologically parallel and denote this relation by e op f.A set of opposite edges, within the same ring eventually forming a strip of adjacent rings, is called an opposite edge strip ops, which is a quasi orthogonal cut (qoc).The length of ops is maximal irrespective of the starting edge.Let

( )
, m G c be the number of ops strips of length c.
The physico-chemical properties of chemical compounds are often modeled by means of molecular graph based structure descriptors, known as topological indices [2], [3].The Wiener index is the first distance based topological index [4].The Wiener index of a graph G is defined as The Cluj-Ilumenau index [6] is defined with the help of first and second derivative of Omega polynomial at , , , The Omega index is defined as

Discussion and Main Result
Polycylic Aromatic Hydorcarbons ( t PAH ) are a group of more than 100 different chemicals, these are formed during the incomplete burning of coal, oil, gas, garbage or other substances.To obtain the required result, we used the Cut Method [23]- [25].We calculated the  , i e e C ∀ ∈ .Also from Figure 2 one can notice that the number of repetition of these qoc and the number of repetition of t C is three times.i.e.From this, we obtain that ( ) ( ) This gives that the Omega polynomial of the Polycyclic aromatic hydrocarbons t PAH for all non-negative integer number t is equal to

1 ]
finite connected graph, where V and E are the sets of vertices and edges, respectively.The distance between two vertices u and v in a graph G is the length of the shortest path connecting them, it is denoted by ( ) , d u v .Two edges e uv = and f yz = in graph G are said to be codistant if they satisfy the following condition [ t PAH are usually found as a mixture containing two or more of these compounds.For further information and results on t PAH and other molecular graphs and nano-structures, we refer [7]-[22].In this section, we computed the Omega and Cluj-Ilumenau index of Polycyclic aromatic hydrocarbons t PAH .Theorem 1.Consider the graph of Polycyclic aromatic hydrocarbons t PAH , then we have the following( )

Figure 1 .
Figure 1.General representation of polycyclic aromatic hydrocarbons t PAH .

Figure 2 .
Figure 2. A quasi orthogonal cuts strips on polycyclic aromatic hydrocarbons t PAH .
help of above polynomial we will investigate the Cluj-Ilmenau and Omega indices of Polycyclic aromatic hydrocarbons t PAH .

(
Consider the general representation of the Polycyclic aromatic hydrocarbons