A GAP PROGRAM FOR COMPUTING THE HOSOYA POLYNOMIAL AND WIENER INDEX OF NANO STRUCTURES

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1 Digest Jornal of Nanomaterials and Biostrctres Vol, No, Jne 009, p A GAP PROGRAM FOR COMPUTING THE HOSOYA POLYNOMIAL AND WIENER INDEX OF NANO STRUCTURES A R ASHRAFI, MODJTABA GHORBANI Institte of Nanoscience and Nanotechnology, Uniersity of Kashan, Kashan , I R Iran The Wiener index W(G) is defined as the sm of distances between all pairs of ertices of the G In this paper, we present a GAP program to Compting the Hosoya polynomial and the Wiener index of eery graph We also rn this program to compte the Hosoya polynomial and the Wiener index of carbon nanocone CNC [n] for n=,,,8 (Receied April 30, 009; accepted May, 009) Keywords: Hosoya polynomial, Wiener index, Carbon nanocones CNC [n] Introdction Mathematical calclations are absoltely necessary to explore important concepts in chemistry Mathematical chemistry is a branch of theoretical chemistry for discssion and prediction of the moleclar strctre sing mathematical methods withot necessarily referring to qantm mechanics Chemical graph theory is an important tool for stdying moleclar strctres This theory had an important effect on the deelopment of the chemical sciences In the past years, nanostrctres inoling carbon hae been the focs of an intense research actiity which is drien to a large extent by the qest for new materials with specific applications Let G be a simple moleclar graph withot directed and mltiple edges and withot loops, the ertex and edge-sets of which are represented by V(G) and E(G), respectiely If x and y are two ertices of G then d(x,y) denotes the length of a minimal path connecting x and y A topological index for G is a nmeric qantity that is inariant nder atomorphisms of G A distance-conting polynomial was introdced by Hosoya k as H(G,x) = d(g,k)x The Wiener index of a graph G, named after the chemist k {x,y} V(G) G Harold Wiener, who considered it in connection with paraffin boiling points, is gien by W(G) = d (x,y), where d G denotes the distance in G Besides its prely graphtheoretic ale, the Wiener index has interesting applications in chemistry We qote[3], which gies an extensie smmary on the arios works, and refer to [] for frther information on the chemical applications In connection with certain inestigations in mathematical chemistry, Schltz [5] considered a graph inariant that he called moleclar topological index and whose essential part is the Schltz index S, Athor to whom correspondence shold be addressed( Modjtabaghorbani@gmail com)

2 390 S(G) = ( δ +δ )d(, ) The modified Schltz index is defined as: * S(G) = ( δ δ ) By aboe eqations, it is easy to constrct graph polynomials haing the property that their first deriaties at x = are eqal to the Schltz and modified Schltz indices These polynomials are S(G, x) = ( δ +δ )x and S(G,x) = ( δ δ )x (,) respectiely * d The most important works on compting topological indices of nanostrctres were done by Didea and his co-athors 6- Ashrafi contined this program to calclate the Wiener index of some other nanostrctres -7 In some research papers the Wiener, hyper Wiener and Szeged indices of nanotbes and nanotori are compted We encorage the reader to conslt 8-0 and references therein for backgrond material as well as basic comptational techniqes In this paper, we contine this program to compte the Hosoya Polynomial and the Wiener index of carbon nano cone CNC [n], Figre Or notation is standard and mainly taken from standard books of graph theory and the books of Trinajestic Main reslts and discssion Let D i ( D i {0,,} ) be the nmber of paths of length i in wiener matrix So, the n i n Hosoya polynomial and the Wiener index are H(G,x) = Dix and W(G) = id i, i= i= respectiely Now sppose S n and K n denoted the star graph and complete graph, respectiely n(n ) Theorem W(G) E, with eqality if and only if G K n n Proof for eery i E(G) = D D i and so, idi id, which completes the i= i= n(n ) first part of or theorem For second part, W(G) = E if and only if for eery i and j ( i, j n ), D i = D j if and only if G K n n Theorem W(G), with eqality if and only if G Kn n(n ) Proof Becase for eery i, i, then W(G) and this complete first part of n(n ) theorem For second part, W(G) = if and only if for eery i, D i = if and only if G K n Theorem3 Let G be a fllerene graph The Schltz and polynomial of G are as follow:

3 39 )S(G,x) = ( δ +δ )x = 3 x = 6H(G,x), {,} V(G) {,} V(G) )S(G) = ( δ +δ )d(, ) = 3 d(, ) = 6 W(G), {,} V(G) {,} V(G) 9 3)S (G,x) = ( δ δ )x = x = 9H(G,x), * d(, ) * 9 )S (G) = ( δ δ ) = = 9W(G) {,} V(G) {,} V(G) Proof We know fllerenes are 3- reglar graphs Ths, for complete the proof V(G), δ = 3 and this Now we compte the Hosoya polynomial and the Wiener Index of any Connected graph To do this, we first draw the molecle by HyperChem and then compte the distance matrix of the moleclar graph by TopoClj Finally, we prepare a GAP 3 program for compting the Hosoya polynomial and the Wiener Index of any graph In Table we compte the Hosoya polynomial and the Wiener Index of Carbon nanocones CNC [n] for n=,,,8 Table Vales of W(CNC [n]) and H(CNC [n]), for n 8 n Hosoya Polynomial of CNC [n] Wiener Index +x 0 6+0x+60x +6x 3 +x +x x+6x +08x 3 +0x +08x 5 +60x 6 +x 7 +60x 8 +x x+36x +x 3 +9x +58x 5 +50x 6 +80x x 8 +96x 9 +96x 0 + 0x +6x + x x+56x 5 +7x x +968x 5 +0x 6 +00x x 8 +9x 9 +76x 0 +66x + 56x + 38x 3 +x +0x 5 +6x 6 +x x+76x 6 +07x 3 +3x +58x 5 +67x x x x 9 +68x 0 +5x +3 8x +x x +680x 5 +9x x 7 +6x 8 +0x 9 +6x 0 +x x+0x +008x 3 +50x +3008x x x x 8 +5x 9 +x 0 +7x x +080x x +3560x 5 +36x x 7 +3x x 9 +60x 0 +80x +980x +70x 3 +5x +336x 5 +6x 6 +0x 7 +6x 8 +x x+796x +58x 3 +39x +398x 5 +80x x x x x x x +6056x 3 +59x +570x 5 +50x x 7 +5x x x 0 +99x +508x +080x 3 +67x +3x x 6 +70x 7 +5x x 9 +6x 30 +0x 3 +6x 3 +x

4 39 Fig The graph of Carbon Nanocones CNC [n] A GAP Program for Compting The Hosoya Polynomial And Wiener Index f:=fnction(m) local h,i,j,g,gg,a; h:=[];g:=[];gg:=[]; for i in M do for j in i do Add(h,j); Sort(h); for i in h do for j in h do if j=i then Add(g,j); fi; AddSet(gg,g);g:=[]; for i in [Length(gg)-] do Print(Length(gg[i]),"x"); Print(gg[i][]);Print("+"); a:=length(gg); Print(Length(gg[a]),"x"); Print(gg[a][],"\n"); Print("************************","\n"); Print("\n"); retrn; end;

5 393 References [] H Hosoya, Discrete Appl Math, 9, 39(988) [] H Wiener, J Am Chem Soc 69, 7 (97) [3] A A Dobrynin, R Entringer, I Gtman, Acta Appl Math 66, (00) [] N Trinajstic, Chemical Graph Theory, CRC Press, Boca Raton, FL (99) [5] H P Schltz, J Chem Inf Compt Sci 9, 7 (989) [6] M V Didea, M Stef, B Pâr and P E John, Croat Chem Acta 77, (00) [7] M V Didea, B Par and E C Kirby, MATCH Commn Math Compt Chem 7, 53 (003) [8] M V Didea, Bll Chem Soc Japan 75, 87 (00) [9] M V Didea, MATCH Commn Math Compt Chem 5, 09 (00) [0] M V Didea, P E John, MATCH Commn Math Compt Chem, 03 (00) [] M V Didea, E C Kirby, Fllerene Sci Technol, 9, 5 (00) [] S Yosefi, AR Ashrafi, J Math Chem, 03 (007) [3] A Iranmanesh, B Soleimani, MATCH Commn Math ComptChem, 57, 5 (007) [] A R Ashrafi, S Yosefi, Nanoscale Res Lett, 0 (007) [5] S Yosefi, AR Ashrafi, MATCH Commn Math Compt Chem 56, 69 (006) [6] A R Ashrafi, S Yosefi, MATCH Commn Math Compt Chem 57, 03 (007) [7] A R Ashrafi, B Manoochehrian and H Yosefi-Azari, Util Math 7, 97 (006) [8] A A Dobrynin, I Gtman, S Klažar, P Zigert, Acta Appl Math 7, 7 (00) [9] M Ghorbani and A R Ashrafi, J Compt Theor Nanosci 3, 803 (006) [0] A R Ashrafi, M Ghorbani, MATCH Commn Math Compt Chem, 60, 359 (008) [] HyperChem package Release 75 for Windows, Hypercbe Inc, 5 NW th Street, Gainesille, Florida 360, USA 00 [] M V Didea, O Urs, Cs L Nagy, TOPOCLUJ, Babes-Bolyai Uniersity, Clj 00 [3] M Schönert, HU Besche, Th Breer, F Celler, B Eick, V Felsch, A Hlpke, J Mnich, W Nickel, G Pfeiffer, U Polis, H Theißen and A Niemeyer, GAP, Grops, Algorithms and Programming, Lehrsthl De für Mathematik, RWTH, Aachen, 995