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

Computational Chemistry
Vol.04 No.04(2016), Article ID:71297,6 pages
10.4236/cc.2016.44009

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

M. R. Rajesh Kanna1, R. Pradeep Kumar2, Muhammad Kamran Jamil3, Mohammad Reza Farahani4

1Department of Mathematics, Maharani’s Science College for Women, Mysore, India

2Department of Mathematics, The National Institute of Engineering, Mysuru, India

3Department of Mathematics, Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, Lahore, Pakistan

4Department of Applied Mathematics, Iran University of Science and Technology (IUST), Tehran, Iran Copyright © 2016 by authors and Scientific Research Publishing Inc.   Received: August 11, 2016; Accepted: October 16, 2016; Published: October 19, 2016

ABSTRACT

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 for all integer number k.

Keywords:

Molecular Graph, Hydrocarbons, Topological Indices 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 .

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  ,  . The Wiener index is the first distance based topological index  . The Wiener index of a graph G is defined as .

M. V. Diudea introduced the Omega Polynomial for counting ops strips in graph G .

First derivative of Omega polynomial at equals the size of the graph G, i.e. .

The Cluj-Ilumenau index  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  -  . 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  -  . 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

Cite this paper

Kanna, M.R.R., Kumar, R.P., Jamil, M.K. and Farahani, M.R. (2016) Omega and Cluj-Ilmenau Indices of Hydrocarbon Molecules “Polycyclic Aromatic Hydrocarbons PAHk”. Computational Chemistry, 4, 91-96. http://dx.doi.org/10.4236/cc.2016.44009

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