Int. J. Contemp. Math. Sciences, Vol. 6, 20, no. 5, 25-220 Second-Order and Third-Order Connectivity Indices of an Infinite Family of Dendrimer Nanostars Jianguang Yang, Fangli Xia and Shubo Chen Department of Mathematics, Hunan City University, Yiyang Hunan, 43000, P.R. China hunancity@gmail.com Abstract The m-order connectivity index m χ(g) of a molecular graph G is the sum of the weights (d i d i2 d im+ ) 2, where d i d i2 d im+ runs over all paths of length m in G and d i denotes the degree of vertex v i. Dendrimer is a polimer molecule with a distinctive structure that resembles the crown of a tree. In this paper we compute second-order and third-order connectivity indices of an infinite family of dendrimer nanostars. Mathematics Subject Classification: 05C05, 05C2 Keywords: Connectivity index, second-order connectivity index, thirdorder connectivity index, dendrimer nanostar Introduction In 975, Randić introduced a molecular structure-descriptor in his study of alkanes[] which he called the branching index, and is now called the the Randić index or the connectivity index. The Randić index has been closely correlated with many chemical properties (see [2]). Let G =(V,E) be a simple graph with vertex set {v,v 2,,v n }. For any two vertices v i,v j V (G) with i<j, we will use the symbol ij to denote the edge e = v i v j.forv i V, the degree of v i, written by d i, is the number of edges incident with v i. For an integer m, the m order connectivity index of a graph G is defined as m χ(g) = i i 2 i m+ di d i2 d im+ ()
26 J. Yang, F. Xia and S. Chen where i i 2 i m+ runs over all paths( that is i s i t for s<t m +) of length m of G. The higher order connectivity indices are of great interest in molecular graph theory ([3-4]) and some of their mathematical properties have been reported in [5-7]. In particular, the second order( or simply write as 2-order) connectivity and the third order( or simply write as 3-order) connectivity indices are defined as 2 χ(g) = di d i2 d i3, i i 2 i 3 3 χ(g) = di d i2 d i3 d i4 (2) i i 2 i 3 i 4 N 2 N N 0 2 e 0 N 0 N 3 N 4 Figure. The nanostar dendrimer NS[2] Dendrimer is a synthetic 3-dimentional macromolecule that is prepared in a step-wise fashion from simple branched monomer units. The nanostar dendrimer is part of a new group of macromolecules that appear to be photon funnels just like artificial antennas[8-0]. During the past several years, there are many papers dealing with the topological indices of dendrimer nanostars, the readers may consult [0-4] and references cited therein. We shall consider the 2-order connectivity index and 3-order connectivity indices of an infinite family of dendrimer nanostars NS[n] described in Figure [4].
Second-order and third-order connectivity indices of dendrimer nanostars 27 2 The 2-order and 3-order connectivity indices of NS[n] In the following, we shall compute the 2-order connectivity index and 3-order connectivity index for the dendrimer nanostar NS[n] as shown in Figure. Theorem 2. Let NS[n] be the dendrimer nanostar as shown in Figure. Then ( ) ( 2 χ(ns[n]) = )( 9 23 2 + 3 3 + 9 2 + 70 3 2 9 2 n+ 4 ) Proof. Firstly, we compute 2 χ(ns[]). Let d ijk denotes the number of 2-path whose three consecutive vertices are of degree i, j, k, resp. Similarly, we use ijk to mean d ijk in s th stage. Particularly, ijk = d(s) kji. It is ease to see that d () 222 =48,d () 223 =32,d () 232 =22,d () 233 =60,d () 323 =0,d () 333 =44. Therefore, we have that 2 χ(ns[]) = 48 2 2 2 + 32 2 2 3 + 22 2 3 2 + 60 2 3 3 + 0 3 2 3 + 44 3 3 3 = 9 (23 2 + 25 3) Secondly, we establish the relation between 2 χ(ns[s]) and 2 χ(ns[s ]) for s 2. By simple reduction, we are ready to have, 222 = d (s ) 222 +8 2 s, 223 = d (s ) 223 +20 2 s, 232 = d (s ) 232 +2 2 s, 233 = d (s ) 233 +28 2 s, 323 = d (s ) 323 +2 2 s 333 = d (s ) 323 +22 2 s. and for any (i, j, k) (222), (223), (232), (233), (323), (333) we have 0or ijk = d(s ) rst = = d () ijk for s =2, 3, 4,,n. Thus, 2 χ(ns[n]) = 2 χ(ns[n ]) + 8 2n 2 2 2 + 20 2n 2 2 3 + 2 2n 2 3 2 + 28 2n 2 3 3 + 2 2n 3 2 3 + 22 2n 3 3 3 = 2 χ(ns[n ]) + ( 9 2 + 70 3)2 n 2 9 From above recursion formula, we have ijk =
28 J. Yang, F. Xia and S. Chen 2 χ(ns[n]) = 2 χ(ns[n ]) + ( 43 2 + 46 3 )2 n 6 9 = 2 χ(ns[n 2]) + ( 43 2 + 46 3)(2 n +2 n ) 6 9. = 2 χ(ns[]) + ( 9 2 + 70 3) ( 2 n +2 n + +2 2) ( 2 9 ) ( = )( 9 23 2 + 25 3 + 9 2 + 70 3 2 9 2 n+ 4 ) This completes the proof. Theorem 2.2 Let NS[n] be the dendrimer nanostar as shown in Figure. Then ) ( )( 3 χ(ns[n]) = 9( 222 + 94 6 + 9 99 + 44 6 2 n+ 4 ) Proof. Let d ijkl denotes the number of 3-paths whose three consecutive vertices are of degree i, j, k, l, resp. At the same time, we use ijkl to mean d ijkl in s th stage. Obviously, ijkl = d(s) lkji. Similar to the discussion way in Theorem 2., at the first, we compute 3 χ(ns[]). It is easy to see that d () 2222 =32,d () 2223 =32,d () 2232 =32,d () 2233 =32,d () 2323 =2,d () 2332 =24,d () 2333 = 64,d () 3232 =2,d () 3233 =28,d () 3333 = 48. Thus, 3 χ(ns[]) = 32 2 2 2 2 + 32 32 2 2 2 3 + 32 2 2 3 2 + 2 2 3 3 + 64 + 2 2 3 3 3 + 28 3 2 3 2 + 48 3 2 3 3 + 3 3 3 3 = (222 + 94 6) 9 Secondly, we compute 3 χ(ns[n]). The relations between ijkl and d(s ) ijkl for s 2 are 2222 = d (s ) 2222 +2 2 s, 2223 = d (s ) 2223 +2 2 s, 2232 = d (s ) 2232 +20 2 s, 2233 = d (s ) 2233 +20 2 s, 2332 = d (s ) 2332 +8 2 s, 2333 = d (s ) 3233 +32 2 s 3233 = d (s ) 3233 +4 2 s 3333 = d (s ) 3233 +24 2 s. 24 2 3 3 2 For any (i, j, k, l) (2222), (2223), (2232), (2233), (2332), (2333), (3233), (3333), ijkl =0ord(s) ijkl = d(s ) ijkl = = d () ijkl for s =2, 3, 4,,n. Thus,
Second-order and third-order connectivity indices of dendrimer nanostars 29 3 χ(ns[n]) = 3 χ(ns[n ]) + 2 2n + 2 2n 4 2 + 20 2n 6 2 + 20 2n + 8 2n 6 6 6 + 32 2n 3 + 4 2n + 8 2n 6 6 3 + 24 2n 6 9 = 3 χ(ns[n 2]) + 2n (99 + 44 6) 9. = 3 χ(ns[]) + (99 + 44 6) ( 2 n +2 n + +2 2) ( 9 ) ( )( = 9 222 + 94 6 + 9 99 + 44 6 2 n+ 4 ) This completes the proof. Acknowledgements. Project supported by the Research Foundation of Education Bureau of Hunan Province of China(Grant No: 0B05) and the Science and Technology of Hunan City University(Grant No. 200xj006). References [] M. Randić, On characterization of molecular branching, J. Amer. Chem. Soc. 97(975), 6609-665. [2] L. B. Kier and L. H. Hall, Molecular Connectivity in Chemistry and Drug Research, Academic Press, San Francisco, 976. [3] L. B. Kier, W. J. Murray, M. Randić and L. H. Hall, Molecular connectivity V: connectivity series concept applied to density, J. Pharm. Sci. 65(976), 226-230. [4] V. K. Singh, V. P. Tewary, D. K. Gupta and A. K. Srivastava, Calculation of heat of formation: molecular connectivity and IOC-ω technicque, a comparative study, Tetrahedron 40(984), 2859-2863. [5] O. Araujo and J.A. de la Peña, The connectivity index of a weighted graph, Linear. Alg. Appl. 283(998), 7-77. [6] J. Rada and O. Araujo, Higher order connectivity index of starlike trees, Discr. Appl. Math. 9(2002), 287-295. [7] M. Randić, Representation of molecular graphs by basic graphs, J. Chem. Inf. Comput. Sci. 32(992), 57-69. [8] M.B.Ahmadi, M. Sadeghimehr. Second-order connectivity index of an infinite class of dendrimer nanostars. Digest Journal of Nanomaterials and Biostructures. 4(2009), 639-643. [9] M. B. Ahmadi, M. Seif, The Merrifield-Simmons index of an infinite class of dendrimer nanostars, Digest Journal of Nanomaterials and Biostructures. 5(200), 335-338.
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