Omega and Cluj-Ilmenau Indices of Hydrocarbon Molecules “Polycyclic Aromatic Hydrocarbons PAHk” ()
1. Introduction
Let 
be a simple 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
. Two edges 
and 
in graph G are said to be codistant if they satisfy the following condition [1]
.
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 equiva- lence relation. A graph G in which co is an equivalence relation is called co-graph, and the subset of edges 
is called an orthogonal cut (oc) of G, also the edge set 
can be written as the union of disjoint orthogonal cuts, i.e.
.
Let 
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 
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
.
M. V. Diudea introduced the Omega Polynomial 
for counting ops strips in graph G [5]
.
First derivative of Omega polynomial at 
equals the size of the graph G, i.e.
.
The Cluj-Ilumenau index [6] is defined with the help of first and second derivative of Omega polynomial at 
as
.
The Omega index is defined as
![]()
.
2. Discussion and Main Result
Polycylic Aromatic Hydorcarbons (![]()
) are a group of more than 100 different chemicals, these are formed during the incomplete burning of coal, oil, gas, garbage or other substances. ![]()
are usually found as a mixture containing two or more of these compounds. For further information and results on ![]()
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![]()
.
Theorem 1. Consider the graph of Polycyclic aromatic hydrocarbons![]()
, then we have the following
![]()
![]()
.
Proof Consider the general representation of the Polycyclic aromatic hydrocarbons ![]()
as shown in Figure 1. The structure of ![]()
contain ![]()
atoms/vertices and ![]()
bonds/edges.
To obtain the required result, we used the Cut Method [23] - [25] . We calculated the ![]()
for all opposite edge strips. From Figure 2, it is clear that there are ![]()
distinct cases of qoc strips for ![]()
and the graph of Polycyclic aromatic hydro- cabons’s graph is a co-graph. The size of a qoc strip is ![]()
for
![]()
and![]()
. Because there are ![]()
co-distant edges with
![]()
Figure 1. General representation of polycyclic aromatic hydro- carbons![]()
.
![]()
Figure 2. A quasi orthogonal cuts strips on polycyclic aro- matic hydrocarbons![]()
.
![]()
. Also from Figure 2 one can notice that the number of repetition of these qoc stips ![]()
is six ![]()
and the number of repetition of ![]()
is three times. i.e.
・ For![]()
, ![]()
and ![]()
・ For all![]()
, ![]()
and ![]()
・ For![]()
, ![]()
and ![]()
From this, we obtain that
![]()
.
This gives that the Omega polynomial of the Polycyclic aromatic hydrocarbons ![]()
for all non-negative integer number t is equal to
![]()
.
Now with the help of above polynomial we will investigate the Cluj-Ilmenau and Omega indices of Polycyclic aromatic hydrocarbons![]()
.
![]()
![]()
As
![]()