ON CERTAIN TOPOLOGICAL INDICES OF BENZENOID COMPOUNDS

Drug discovery is mainly the result of chance discovery and massive screening of large corporate libraries of synthesized or naturally-occurring compounds. Computer aided drug design is an approach to rational drug design made possible by the recent advances in computational chemistry in various fields of chemistry, such as molecular graphics, molecular mechanics, quantum chemistry, molecular dynamics, library searching, prediction of physical, chemical, and biological properties. The structure of a chemical compound can be represented by a graph whose vertex and edge specify the atom and bonds respectively. Topological indices are designed basically by transforming a molecular graph into a number. A topological index is a numeric quantity of a molecule that is mathematically derived from the structural graph of a molecule. In this paper we compute certain topological indices of pyrene molecular graph. The topological indices are used in quantitative structure-property relationships (QSPR) and quantitative structure-activity relationships (QSAR) studies. The new multiplicative versions of M 1 ( G ) and M 2 ( G ) indices, denoted by PM 1 ( G ) and PM 2 ( G ) (respectively), were first defined by Ghorbani and Azimi [7]. These indices are obtained for TiO 2 Nanotubes in [8] and defined as follows.


INTRODUCTION
Let be a connected molecular graph. The distance between two distinct vertices denoted by is the length of a shortest path in Topological indices are designed basically by transforming a molecular graph into a number. The concept of topological index came from work done by Harold Wiener in 1947 while he was working on boiling point of paraffin [1]. The Wiener index of G is defined as Two important topological indices introduced about forty years ago by Ivan Gutman and Trinajstic [2] are the first zagreb index M1(G) and the second zagreb index M2(G) which are defined as where denotes the degree of The degree distance was introduced by Dobrynin and Kochetova [3] and the weighted version of the Wiener index [4] is introduced by Gutman in 1994. The degree distance of G and Gutman index of G respectively defined and denoted by I S S N 2 3 2 1 -8 0 7 X V o l u m e 1 3 N u m b e r 8 J o u r n a l o f A d v a n c e s i n c h e m i s t r y 6407 | P a g e F e b r u a r y 2 0 1 6 w w w . c i r w o r l d . c o m

MATERIAL AND METHODS
Pyrene is a polycyclic aromatic hydrocarbon consisting of four fused benzene rings, resulting in a flat aromatic system. The chemical formula is C16H10. This colorless solid is the smallest peri-fused PAH. It is one of a group of chemicals called polycyclic aromatic hydrocarbons, PAHs for short. PAHs are often found together in groups of two or more. They can exist in over 100 different combinations but the most common are treated as a group of 15. PAHs are found naturally in the environment but they can also be man-made. Pyrene is colorless crystal-like solid but can also look yellow. PAHs are created when products like coal, oil, gas, and garbage is burned but the burning process is not complete.
Not much information is available on the individual chemicals within the PAH group. Most of the information available is for the PAH group as a whole. Information specific to pyrene is included in this fact sheet when available. Most of the PAHs are used to conduct research. Like most PAHs, pyrene is used to make dyes, plastics and pesticides. It has also been used to make other PAHs called benzo(a)pyrene, benzo(e)pyrene and 2H-benzo(cf)pyrene-2. See Figure 1.

Fig 1: (i) pyrene ; (ii) benzo (a) pyrene ; (iii) benzo (e) pyrene ; (iv) 2H-benzo (cf) pyrene-2
Benzo[a]pyrene is a polycyclic aromatic hydrocarbon found in coal tar with the formula C20H12. Its metabolites are mutagenic and highly carcinogenic, and it is listed as a Group 1 carcinogen by the International Agency for Research on Cancer (IARC). Benzo[e]pyrene is a polycyclic aromatic hydrocarbon with the chemical formula C20H12. It is listed as a Group 3 carcinogen by the IARC. Resistomycin (6CI), Antibiotic A 3733A, Geliomycin, Itamycin are the other names of 2H-Benzo[cd]pyrene-2.

RESULTS
Pyrene PY (n) has 2n 2 + 4n vertices and 3n 2 + 4n − 1 edges and it is bipartite as this does not contain odd cycle. See Figure. 2. The bondage number, packing of pyrene and pyrene torus is discussed in [5].

Zagreb Indices
The first and second Zagreb indices were first introduced by I. Gutman and N. Trinajstic in 1972. It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. Recently, the first and second multiple Zagreb indices of a graph G were introduced by Ghorbani and Azimi in 2012. The Zagreb indices were first appeared in the topological formula for the total π-energy of conjugated molecules that has been derived in 1972 by Gutman and Trinajstic [2]. These indices have also been used as branching indices [6] and have found applications in QSPR and QSAR studies.
The first Zagreb index M1(G) and the second Zagreb index M2(G) of a molecular graph G are respectively defined as I S S N 2 3 2 1 -8 0 7 X V o l u m e 1 3 N u m b e r 8 J o u r n a l o f A d v a n c e s i n c h e m i s t r y 6408 | P a g e F e b r u a r y 2 0 1 6 w w w . c i r w o r l d . c o m The new multiplicative versions of M1(G) and M2(G) indices, denoted by PM1(G) and PM2(G) (respectively), were first defined by Ghorbani and Azimi [7]. These indices are obtained for TiO2Nanotubes in [8] and defined as follows.

ABC Index, GA index and Randi´c index
The first geometric-arithmetic (GA) index was proposed by Vukiˇcevi´c and Furtula [9]. It has been demonstrated, on the example of octane isomers, that GA index is well-correlated with a variety of physico-chemical properties [9]. The details about mathematical properties of the GA indices and their applications in QSPR and QSAR can be found in the survey [10].
The atom-bond connectivity ABC index of graph G is proposed by Estrada et al. in [12] which is abbreviated as ABC(G) and defined as The ABC index provides a good model for the stability of linear and branched alkanes as well as the strain energy of cycloalkanes [3,12]. Due to its physico-chemical applicability, the ABC index has attracted significant attention from researchers in recent years and many mathematical properties of this index were reported in [12,13,14].
For a graph G = (V,E), the general Randi´c index Rα(G) is defined as the sum of (du × dv) α over all edges e = uv of G.
The Randi´c index was introduced in [15] by Randi´c in 1975 with α = −0.5 as one of the many graph-theoretical parameters derived from the graph underlying some molecule. This index has been closely correlated with many chemical properties. The general Randi´c index was proposed by Bollob´as and Erd¨os [16], and Amic et al. [17], independently, in 1998. Then it has been extensively studied by both mathematicians and theoretical chemists [18]. Many important mathematical properties have been established [19]. The following Figure 4 Explains the comparative study of the above derived topological indices.

CONCLUSION
In this paper we have obtained some of the degree based topological indices of pyrene molecular graph. The other topological indices are under investigation.