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

Show more

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

References

[1] 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.

[2] Diudea, M.V. (2001) Wiener Index of Dendrimers. NOVA, New York.

[3] Trinjastic, N. (1992) Chemical Graph Theory. CRC Press, Boca Raton.

[4] Wiener, H. (1947) Structural Determination of Paraffin Boiling Points. Journal of the American Chemical Society, 69, 17-20. http://dx.doi.org/10.1021/ja01193a005

[5] Diudea, M.V. (2006) Omega Polynomial. Carpathian Journal of Mathematics, 22, 43-47.
http://carpathian.ubm.ro/?m=past_issues&issueno=Vol

[6] Diudea, M.V. (2010) Counting Polynomials and Related Indices by Edge Cutting Procedures. MATCH Communications in Mathematical and in Computer Chemistry, 64, 569.

[7] 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.2016.1.00023

[8] 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.

[9] 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.

[10] 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.

[11] 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.

[12] Jamil, 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.

[13] 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.2016.1.00019

[14] Liu, J.B., Wang, C., Wang, S. and Wei, B. (Submitted) Zagreb Indices and Multiplicative Zagreb Indices of Eulerian Graphs.

[15] Wang, C., Wang, S. and Wei, B. (2016) Cacti with Extremal PI Index. Transactions on Combinatorics, 5, 1-8.

[16] Wang, S. and Wei, B. Padmakar-Ivan Indices of K-Trees. (Submitted Paper).

[17] Wang, 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.2016.714140

[18] Wang, 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/s1793830916500403

[19] 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.20160503.17

[20] Wang, 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.

[21] 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.2014.08.017

[22] 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.

[23] Farahani, M.R. (2013) Using the Cut Method to Computing Edge Version of Co-PI Index of Circumcoronene Series of Benzenoid . Pacific Journal of Applied Mathematics, 5, 65-72.

[24] John, 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/s10910-006-9100-2

[25] Klavzar, S. (2008) A Bird’s Eye View of the Cut Method and a Survey of Its Applications in Chemical Graph Theory. MATCH Communications in Mathematical and in Computer C, 60, 255-274.