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Molecular descriptors of benzenoid systems |

Nazeran Idrees 1. Department of Mathematics, Government College University Faisalabad, Faisalabad, 38000, Punjab, Pakistan Recebido em 31/05/2016 Endereço para correspondência
*e-mail: nazeranjawwad@gmail.com RESUMO
Molecular descriptors are being widely used in QSAR/QSPR studies in chemistry and drug designing as well as modeling of compounds. Different topological descriptors have been formulated to investigate the physio chemical properties and chemical reactivity of compounds. In this article we gave exact relations for first and second Zagreb index, hyper Zagreb index, multiplicative Zagreb indices as well as first and second Zagreb polynomials for some benzenoid systems. Palavras-chave: benzenoid hydrocarbons; chemical graphs; topological indices.
Benzenoid hydrocarbons are present naturally in our environment, minerals, food and are also produced as byproducts in certain reaction and have large scale applications in chemical industry. Apart from their extensive use, benzenoid hydrocarbons are carcinogenic and pollutants. Benzenoid systems are actually hydrogen depleted benzenoid hydrocarbons. Benzenoid systems are planar, consist of regular hexagons which do no overlap and have no hole as shown in Figure 1.
Figure 1. An example of benzenoid system(a) and a non - benzenoid system (b)
Let Zagreb Indices are one of the oldest known topological invariants which first appeared as terms in a formula for analysis of π-electron energy and they grow with the branching of chemical graphs. In 2013, Shirdel Ghorbani and Azimi defined first multiple Zagreb index The first Zagreb polynomial These new variants of Zagreb indices have been extensively studied recently.
Consider the graph Z and there are two hexagons in each row of the system. _{n}Z has 8_{n}n + 2 vertices and 10n + 1 edges.
Figure 2. Graph of zigzag benzenoid system Z_{n}
We divide edge set on the basis of degrees of endpoints of edges of the graph and also compute number of edges in each set of the partition. All vertices have degree either two or three. The edged whose endpoint vertices have degree two can be counted by analyzing the patterns for different values of n. One can observe that at each level we have two such edges, one upside and one downside the chain except the endpoints where we have two more edges with endpoint degree two. Thus having total 2
We compute first Zagreb index, second Zagreb index, hyper-Zagreb index Polynomials
M_{2}(Z) of zigzag benzenoid system _{n},xZ are_{n}Proof. First and second Zagreb polynomial of zigzag benzenoid chain
Consider a benzenoid system in which hexagons are arranged to form a rhombic shape, say, n(n + 2) vertices and 3n^{2} + 4n - 1 edges. Many different benzenoid systems have been studied recently for topological invariants like trapezoid, triangular, circumcoronene and jagged rectangles.
Figure 3. Graph of rhombic benzenoid system with n hexagons along each boundary
Let H be the graph of zigzag benzenoid system
In the next theorem we compute first Zagreb index
Proof. Using edge partition given in Table 2, we compute different variants of Zagreb indices defined in equations 1-5 as
M_{2}(R) of rhombic benzenoid system _{n},xR are_{n}Proof. Now we proceed to compute the first Zagreb polynomial and second Zagreb polynomial of
Different variants of Zagreb indices and Zagreb polynomials are analyzed for two important benzenoid systems using edge partition based on degree of vertices of the edges of the corresponding chemical graphs. We found exact relations of First Zagreb index, second Zagreb index, hyper Zagreb index, multiplicative Zagreb indices as well as Zagreb polynomials for zigzag benzenoid system and rhombic benzenoid system in above theorems, which have not been computed earlier according to the best of our knowledge.
1. Gutman, I.; Cyvin, S. J.; 2. Gutman, I.; Trinajstic, N.; 3. Balaban, A. T.; Motoc, I.; Bonchev, D.; Mekenyan, O.; 4. Shirdel, G. H.; Pour, H. R.; Sayadi, A. M.; 5. Ghorbani, M.; Azimi, N.; 6. Eliasi, M.; Iranmanesh, A.; Gutman, I.; 7. Furtula, B.; Gutman, I.; Dehmer, M.; 8. Gutman, I.; Das, K. C.; 9. Gutman, I.; Furtula, B.; Vukicevic, Z. K.; Popivoda, G.; 10. Gutman, I.; 11. Hayat, S.; Imran, M.; 12. Rada, J.; Cruz, R.; Gutman, I.; |

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