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Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,a
Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,a
Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,a
Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,a
Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,a
Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,a
Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,a
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Theta θ(g,x) and pi π(g,x) polynomials of hexagonal trapezoid system tb,a

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A counting polynomial, called Omega Ω(G,x), was proposed by Diudea. It is defined on the ground of …

A counting polynomial, called Omega Ω(G,x), was proposed by Diudea. It is defined on the ground of
“opposite edge strips” ops. Theta Θ(G,x) and Pi Π(G,x) polynomials can also be calculated by ops
counting. In this paper we compute these counting polynomials for a family of Benzenoid graphs that called
Hexagonal trapezoid system Tb,a.

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  • 1. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013 THETA Θ(G,X) AND PI Π(G,X) POLYNOMIALS OF HEXAGONAL TRAPEZOID SYSTEM TB,A Mohammad Reza Farahani1 1 Department of Applied Mathematics, Iran University of Science and Technology (IUST) Narmak, Tehran, Iran ABSTRACT A counting polynomial, called Omega Ω(G,x), was proposed by Diudea. It is defined on the ground of “opposite edge strips” ops. Theta Θ(G,x) and Pi Π(G,x) polynomials can also be calculated by ops counting. In this paper we compute these counting polynomials for a family of Benzenoid graphs that called Hexagonal trapezoid system Tb,a. KEYWORDS Molecular graph, Benzenoid, Hexagonal trapezoid system, Omega Ω(G,x) polynomial, Theta Θ(G,x) Polynomial, and Pi Π(G,x) polynomial. 1. INTRODUCTION Mathematical calculations are absolutely necessary to explore important concepts in chemistry. Mathematical chemistry is a branch of theoretical chemistry for discussion and prediction of the molecular structure using mathematical methods without necessarily referring to quantum mechanics. In chemical graph theory and in mathematical chemistry, a molecular graph or chemical graph is a representation of the structural formula of a chemical compound in terms of graph theory. A topological index is a numerical value associated with chemical constitution purporting for correlation of chemical structure properties, chemical reactivity or biological activity. Let G(V,E) be a connected molecular graph without multiple edges and loops, with the vertex set V(G) and edge set E(G), and vertices/atoms x,y  V(G). Two edges/bonds e=uv and f=xy of G are called codistant “e co f”, if and only if they obey the following relation: [1, 2] d(v,x)=d(v,y)+1=d(u,x)+1=d(u,y) Relation co is reflexive, that is, e co e holds for any edge e of G; it is also symmetric, if e co f then f co e and in general, relation co is not transitive. If “co” is also transitive, thus an equivalence relation, then G is called a co-graph and the set of edges is C(e):={f  E(G)| e co f}), called an orthogonal cut (denoted by oc) of G. In other words, E(G) being the union of disjoint orthogonal cuts for i≠j & i, j=1, 2, …, k, E G   C  C2  ...  Ck 1  Ck and Ci  Cj  . Klavžar 1 [3] has shown that relation co is a theta Djoković [4], and Winkler [5] relation. Let m(G,c) be the DOI:10.5121/ijcsa.2013.3501 1
  • 2. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013 number of qoc strips of length c (i.e. the number of cut-off edges) in the graph G. Omega Polynomial Ω(G,x), [6-9] for counting qoc strips in G was defined by Diudea as  G , x    m G , c  xc c The summation runs up to the maximum length of qoc strips in G. The first derivative (in x=1) equals the number of edges in the graph   G,1  c m  G, c   c Other counting polynomials “Θ(G,x)” and “Π(G,x)“ was defined as (G, x)   m(G, c)c.x c c (G, x)   m(G, c)c.x E ( G ) c c where m(G,c) is the number of strips of length c. Ω(G,x) and Θ(G,x) polynomials count “equidistant edges” in G and Π(G,x) “non-equidistant edges”. The first derivative (computed at x=1) of these counting polynomials provide interesting topological indices:  '(G,1)  c m(G, c)  c 2  '(G,1)  c m(G, c)  c( E (G)  c) From above equations, one can see that the first derivative of omega polynomial (in x=1), equals the number of edges in the graph G, Ω’(G,1)=|E(G)|. And also, the first derivative of Pi polynomial (in x=1), is equals |E(G)|2-Θ(G,1). In other words, from definition of above counting polynomials (Omega Ω(G,x), Theta Θ(G,x) and Pi Π(G,x)), one can obtain  '(G,1)2   '(G,1)   '(G,1) The aim of this study is to compute Theta Θ(G,x) and Pi Π(G,x) polynomials of Hexagonal trapezoid system graphs. Tb,a. Here our notations are standard and mainly taken from [6-26]. 2. MAIN RESULTS In this section by using Cut Method and Orthogonal Cut Method we compute Theta Θ(G,x) and Pi Π(G,x) polynomials for a family of benzenoid graphs that called Hexagonal trapezoid system Tb,a. Reader can see general representation of this family in Figure 1 and Reference [27]. A hexagonal trapezoid system Tb,a (a  b  ) is a hexagonal system consisting a−b+1 rows of benzenoid chain in which every row has exactly one hexagon less than the immediate row. Theorem 2.1. Consider the hexagonal trapezoid system Tb,a (a  b  ) , Then Theta polynomial of Tb,a is: Θ(Tb,a,x)=4x2+6x3+…+2(a-b+1)xa-b+1+(b+1)xb+1+(b+2)xb+2+…+(a-1)xa-1 +axa+(a+1)xa+1 +(6b2b2)xa-b+2 And Theta index of Tb,a is 2
  • 3. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013 Θ(Tb,a)=a3+b3-2ab2+ 9 2 a2+ 11 2 b2+2ab + 13 2 a- 9 2 b+5 Figure 1: A general representation of the hexagonal trapezoid system Tb,a (a  b  ) In general case the number of vertices/atoms and edges/bonds of the hexagonal trapezoid system 2 a 1 3a 1 3 9 b Tb,a are equal to 2a+1  i 2b1 i =a2−b2+4a+2 and 2a  i 3b1 i = (a 2  b2 )  a   1 for all 2 2 2 a  b . From Figures 1 and By using the Cut Method, one can see that there are two types of edges-cut of hexagonal trapezoid system (we denote the corresponding edges-cut by Ci and Ci) and we have following table. Table 2.1. All quasi-orthogonal cuts of hexagonal trapezoid system Tb,a a  b  . quasi-orthogonal cuts The length of qoc strips The number of qoc strips Ci  i=1,…, a-b i+1 2 Ca b 1 ci  i=1,…,a-b+1 a-b+2 a-i+2 2b 1 Thus, from Table 1, we compute Theta polynomial of hexagonal trapezoid system Tb,a as  Tb,a , x   c m Tb,a , c  .c.x c  C m Tb,a , Ci  .Ci x Ci  m Tb,a , Ca b1  .Ca b1x Cab1  c m Tb,a , ci  .ci .x ci i a b i   m Tb,a , Ci  .Ci x Ci  m Tb,a , Ca b 1  .Ca b1x Cab1  i 1 a b 1  m T i 1 a b a b 1 i 1 b ,a , ci  .ci .x ci i 1  2  i  1 x i 1   2ba  4b  2b2  x a b 2   a  2  i x a  2 i 3
  • 4. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013 =4x2+6x3+…+2(a-b+1)xa-b+1+(b+1)xb+1+(b+2)xb+2+…+(a-1)xa-1 +axa+(a+1)xa+1 +(2b+4b-2b2)xa-b+2   a  1 x a 1  a b 1  a  2  ix i 2 i  x a  2i    2ba  4b  2b 2  x a b  2 Also, Theta index is equal to a b 1  a b 1     2ix i    a  2  i  x a  2i   2ba  4b  2b 2  x a b  2  i 1 | (Tb,a , x) |x1   i 2 x 1 x a b 1 a b 1 i 2 i 1  2  i2  a b  a  2  i a b 2  2b  a  b  2  2  2  i  1    a  1  i    a  1  2b  a  b  2  i 1 2 2 2 2 i 1 a b    3i 2  i  2a  2   a 2  2a  3   a  1  2b  a 2  b2  4  2ab  4a  4b  2 i 1 2  3 a  b  a  b  3 2    a  b      a  1  a  b    a  b    a  b   a 2  2a  3  2 2      +(a2+2a+1)+(2b3+2a2b-4ab2+8ab-8b2+4) =(a3-b3-3a2b+3ab2+ 3 2 a2-3ab+ 3 2 b2 + 1 2 a- 1 2 b)-(a3-2a2b+ab2-a2-b2+2ab+a2 -ab-a+b)+(a3-a2b +2a2-2ab +3a-3b) +(a2+2a+1) +(2b3+2a2b-4ab2+8ab-8b2+4) =(a3-b3-2a2b+2ab2+ 7 2 a2+ 5 2 b2-6ab+ 9 2 a- 9 2 b)+(a2+2a+1)+(2b3+2a2b -4ab2+8ab-8b2+4) =a3+b3-2ab2+ 9 2 a2+ 11 2 b2+2ab + 13 2 a- 9 2 b+5 Theorem 2.2. Pi polynomial Π(Tb,a,x) and Pi index Π(Tb,a) are equal to Π(Tb,a,x)=4xe-2+6xe-3+…+2(a-b+1)xe+b-a-1+(b+1)xe-b-1+(b+2)xe-b-2+…+(a-1)xe-a+1-1 +axe-a+(a+1)xe-a-1+(6b-2b2)xe+b-a-2 and 4
  • 5. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013 Π(Tb,a)= 9 4 a4+ 9 4 b4+ 9 2 a2b2 + 25 2 a3- 5 2 b3 - 23 2 ab2+ 3 2 a2b+ 69 4 a2- 15 4 b2+ 5 2 ab+ 5 2 a+ 11 2 b-4 From Table 2.1 and analogous to Theta polynomial, we have  Tb,a , x    c m Tb,a , c  .c.x E (Tb ,a ) c  c m Tb,a , ci  .ci .x eci  C m Tb,a , Ci  .Ci x eCi  m Tb,a , Ca b1  .Ca b1x eCab1 i  a b 1  m T ci ,i 1  i a b e ci   m Tb,a , Ci  .Ci x eCi  m Tb,a , Ca b 1  .Ca b 1x eCab1 b , a , ci  .ci .x Ci ,i 1 a b 1 a b   a  2  i  x eia2  2 i  1 x ei1   2ba  4b  2b2  xeba2 i 1 i 1 =4x +6x +…+2(a-b+1)x e-2 e-3 +(b+1)xe-b-1+(b+2)xe-b-2+…+(a-1)xe-a+1-1 e+b-a-1 +axe-a+(a+1)xe-a-1+(2b+4b-2b2)xe+b-a-2 Clearly, Pi index Π(Tb,a) can be derived from the definition of Pi polynomial Π(Tb,a,x) by 3 9 b replacing the exponent e (=|E(Tb,a)|) by (a 2  b2 )  a   1 is equal to 2 2 2 2 3 9 b Π(Tb,a)= (Tb,a , x) |x1 =  (a 2  b2 )  a   1   Tb,a    2 2  2 2 = 9 4 a4+ 9 4 b4+ 9 2 a2b2+ 81 4 a2+ b + 27 2 a3- 3 2 b3- 27 2 ab2+ 3 2 a2b+ 3 2 a2+ 3 2 b2 4 + 9 2 ab+9a+b+1 –(a +b3-2ab2+ 9 2 a2+ 11 2 b2+2ab + 13 2 a- 9 2 b+5) 3 = 9 4 a4+ 9 4 b4+ 9 2 a2b2 + 25 2 a3- 5 2 b3 - 23 2 ab2+ 3 2 a2b + 69 4 a2- 15 4 b2+ 5 2 ab+ 5 2 a+ 11 2 b-4 An especial case of this family is triangular benzenoid Gn, that is equivalent with a hexagonal trapezoid system T1,n. It is easy to see that the triangular benzenoid Gn n  has n2+4n+1 vertices and 3 2 n(n+3) edges (see Figure 2). Theorem 2.3. Let Gn= T1,n be the triangular benzenoid. Then  Theta polynomial of Gn is equal to Θ(Gn,x)= Θ(T1,n,x)=6x2+9x3+…+3nxn +(n+5)xn+1  Theta index of Gn is equal to Θ(Gn)= Θ(T1,n)=n3+ 9 2 n2+ 13 2 n +7  Pi polynomial of Gn is equal Π(Gn,x)=6xe-2+9xe-3+…+3nxe-n+(n+5)xe-n-1 where e= 3 2 n(n+3)  Pi index of Gn is Π(Gn)= 9 4 n4+ 25 2 n3+ 93 4 n2- 13 2 n- 7 2 5
  • 6. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013 Figure 2: Graph of triangular benzenoid G7 or T1,7. REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] M.V. Diudea, S. Cigher, A.E. Vizitiu, O. Ursu and P. E. John. Omega polynomial in tubular nanostructures. Croat. Chem. Acta, 79 (3), 445-448. (2006). A.E. Vizitiu, S. Cigher, M.V. Diudea and M.S. Florescu, Omega polynomial in ((4,8) 3) tubular nanostructures. MATCH Commun. Math. Comput. Chem. 57(2), 479-484 (2007). S. Klavžar, MATCH Commun. Math. Comput. Chem. 59, 217 (2008). D.Ž. Djoković, J. Combin. Theory Ser. B, 14, 263 (1973). P.M. Winkler, Discrete Appl. Math. 8, 209 (1984). M.V. Diudea, Omega polynomial. Carpath. J. Math. 22, (2006) 43–47. P.E. John, A.E. Vizitiu, S. Cigher, and M.V. Diudea, MATCH Commun. Math. Comput. Chem. 57, 479 (2007). A.R. Ashrafi, M. Jalali, M. Ghorbani and M.V. Diudea. MATCH, Commun. Math. Comput. Chem. 60 (2008), 905–916 M.V. Diudea and A. Ilić, Note on Omega Polynomial. Carpath. J. Math. 20(1) (2009), 177 – 185. A.R. Ashrafi, M. Ghorbani and M. Jalali, Ind. J. Chem. 47A, 535 (2008). M. Ghorbani, A Note On IPR fullerenes. Digest. J. Nanomater. Bios. 6(2), (2011), 599-602 M.V. Diudea and S. Klavžar. Omega Polynomial Revisited. Acta Chim. Slov. 57, (2010), 565–570 M.V. Diudea. Counting Polynomials in Tori T(4,4)S[c,n]. Acta Chim. Slov. 57, (2010), 551–558 M.V. Diudea, S. Cigher, and P.E. John, Omega and related counting polynomials. MATCH Commun. Math. Comput. Chem. 60, (2008) 237–250. M.V. Diudea, O. Ursu, and Cs.L. Nagy, TOPOCLUJ, Babes-Bolyai University: Cluj. (2002) M.R. Farahani, K. Kato and M.P. Vlad. Omega Polynomials and Cluj-Ilmenau Index of Circumcoronene Series of Benzenoid. Studia Univ. Babes-Bolyai. Chemia 57(3), (2012), 177-182. M.R. Farahani. Computing Θ(G,x) and Π(G,x) Polynomials of an Infinite Family of Benzenoid. Acta Chim. Slov. 59, (2012), 965–968. M.R. Farahani. Omega and Sadhana Polynomials of Circumcoronene Series of Benzenoid. World Applied Sciences Journal. 20(9), 1248-1251, (2012). M.R. Farahani. Sadhana Polynomial and its Index of Hexagonal System Ba,b. International Journal of Computational and Theoretical Chemistry. 1(2), (2013), 7-10. M.R. Farahani. Counting Polynomials of Benzenoid Systems. Proceedings of the Romanian Academy Series B Chemistry 15(3). (2013). In press. 6
  • 7. International Journal on Computational Sciences & Applications (IJCSA) Vol.3, No.5, October 2013 [21] M.R. Farahani. Omega and related counting polynomials of Triangular Benzenoid Gn and linear hexagonal chain LHn. Journal of Chemica Acta, 2, (2013), 43-45. [22] M.R. Farahani. Omega Polynomial of a Benzenoid System. Submitted for publication. (2013). [23] M.R. Farahani. Computing Omega and Sadhana Polynomials of Hexagonal trapezoid system Tb,a. Submitted for publication. (2013). [24] M. Ghorbani, M. Ghazi. Computing Omega and PI polynomials of Graphs. Digest. J. Nanomater. Bios. 5(4), 2010, 843-849 [25] N. Trinajstić, Chemical Graph Theory, CRC Press, Boca Raton, FL, 1992. [26] H. Wiener, Structural determination of the paraffin boiling points. J. Am. Chem. Soc. 69, (1947) 17– 20. [27] Z. Bagheri, A. Mahmiani and O. Khormali. Iranian Journal of Mathematical Sciences and Informatics. 3(1), (2008), 31-39. 7

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