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 PAH
_{k} for all integer number
k.

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

General representation of polycyclic aromatic hydro- carbons<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/1-1710057x40.png" xlink:type="simple"/></inline-formula> A quasi orthogonal cuts strips on polycyclic aro- matic hydrocarbons<inline-formula><inline-graphic xlink:href="http://html.scirp.org/file/1-1710057x42.png" xlink:type="simple"/></inline-formula>

. 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 PAH_{k}”. Computational Chemistry, 4, 91-96. http://dx.doi.org/10.4236/cc.2016.44009

ReferencesKlavzar S. ,et al. (2008)A Bird’s Eye View of the Cut Method and a Survey of Its Applications in Chemical Graph TheoryJohn, P.E., Khadikar, P.V. and Singh, J. (2007) A Method of Computing the PI Index of Benzenoid Hydrocarbons Using Orthogonal Cuts. Journal of Mathematical Chemistry, 42, 27-45. http://dx.doi.org/10.1007/s10910-006-9100-2Farahani M.R. ,et al. (2013)Using the Cut Method to Computing Edge Version of Co-PI Index of Circumcoronene Series of Benzenoid Yan, L., Li, Y., Farahani, M.R., Jamil, M.K. and Zafar, S. (2016) Vertex Version of Co-PI Index of the Polycyclic Armatic Hydrocarbon Systems PAHk. International Journal of Biology, Pharmacy and Allied Sciences, 5, 1244-1253.Wang, S. and Wei, B. (2015) Multiplicative Zagreb Indices of K-Trees. Discrete Applied Mathematics, 180, 168-175. http://dx.doi.org/10.1016/j.dam.2014.08.017Wang, S., Farahani, M.R., Baig, A.Q. and Sajja, W. (2016) The Sadhana Polynomial and the Sadhana Index of Polycyclic Aromatic Hydrocarbons PAHk. Journal of Chemical and Pharmaceutical Research, 8, 526-531.Wang, S., Farahani, M.R., Rajesh Kanna, M.R. and Pradeep Kumar, R. (2016) The Wiener Index and the Hosoya Polynomial of the Jahangir Graphs. Applied and Computational Mathematics, 5, 138-141. http://dx.doi.org/10.11648/j.acm.20160503.17Wang, S. and Wei, B. (2016) Multiplicative Zagreb Indices of Cacti. Discrete Mathematics, Algorithms and Applications, 8, Article ID: 1650040.
http://dx.doi.org/10.1142/s1793830916500403Wang, S.H., Farahani, M.R., Kanna, M.R.R., Kumar, R.P. (2016) Schultz Polynomials and Their Topological Indices of Jahangir Graphs J2,m. Applied Mathematics (Scientific Research Publishing), 7, 1632-1637. http://dx.doi.org/10.4236/am.2016.714140Wang, S. and Wei, B. Padmakar-Ivan Indices of K-Trees. (Submitted Paper).Wang, C., Wang, S. and Wei, B. (2016) Cacti with Extremal PI Index. Transactions on Combinatorics, 5, 1-8.Liu, J.B., Wang, C., Wang, S. and Wei, B. (Submitted) Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs.Jamil, M.K., Farahani, M.R., Imran, M. and Malik, M.A. (2016) Computing Eccentric Version of Second Zagreb Index of Polycyclic Aromatic Hydrocarbons . Applied Mathematics and Nonlinear Sciences, 1, 247-251.
http://dx.doi.org/10.21042/AMNS.2016.1.00019Jamil, M.K., Farahani, M.R. and Rajesh Kanna, M.R. (2016) Fourth Geometric Arithmetic Index of Polycyclic Aromatic Hydrocarbons . The Pharmaceutical and Chemical Journal, 3, 94-99.Farahani, M.R., Rajesh Kanna, M.R., Pradeep Kumar, R. and Wang, S. (2016) The Vertex Szeged Index of Titania Carbon Nanotubes TiO2(m,n). International Journal of Pharmaceutical Sciences and Research, 7, 1000-08.Farahani, M.R., Rehman, H.M., Jamil, M.K. and Lee, D.W. (2016) Vertex Version of PI Index of Polycyclic Aromatic Hydrocarbons . The Pharmaceutical and Chemical, 3, 138-141.Farahani, M.R., Jamil, M.K. and Kanna, M.R.R. (2016) The Multiplicative Zagreb eccentricity Index of Polycyclic Aromatic Hydrocarbons . International Journal of Scientific and Engineering Research, 7, 1132-1135.Farahani, M.R., Jamil, M.K., Kanna, M.R.R. and Kumar, R.P. (2016) Computation on the Fourth Zagreb Index of Polycyclic Aromatic Hydrocarbons . Journal of Chemical and Pharmaceutical Research, 8, 41-45.Alaeiyan, M., Farahani, M.R. and Jamil, M.K. (2016) Computation of the Fifth Geometric Aithmetic Index for Polycyclic Aromatic Hydrocarbons . Applied Mathematics and Nonlinear Sciences, 1, 283-290. http://dx.doi.org/10.21042/AMNS.2016.1.00023Diudea, M.V. (2010) Counting Polynomials and Related Indices by Edge Cutting Procedures. MATCH Communications in Mathematical and in Computer Chemistry, 64, 569.Diudea, M.V. (2006) Omega Polynomial. Carpathian Journal of Mathematics, 22, 43-47.
http://carpathian.ubm.ro/?m=past_issues&issueno=VolWiener, H. (1947) Structural Determination of Paraffin Boiling Points. Journal of the American Chemical Society, 69, 17-20. http://dx.doi.org/10.1021/ja01193a005Trinjastic, N. (1992) Chemical Graph Theory. CRC Press, Boca Raton.Diudea, M.V. (2001) Wiener Index of Dendrimers. NOVA, New York.John, P.E., Vizitiu, A.E., Cigher, S. and Diudea, M.V. (2007) CI Index in Tubular Nanostructures. MATCH Communications in Mathematical and in Computer Chemistry, 57, 479.