WIENER AND DETOUR INDICES OF A NEW TYPE OF NANOSTAR DENDRIMERS

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1 Macedoa Joural of Chemstry ad Chemcal Egeerg, Vol. 8, No., pp (009) MJCCA9 5 ISSN Receved: December, 008 UDC: Accepted: March 5, 009 Orgal scetfc paper WIENER AND DETOUR INDICES OF A NEW TPE OF NANOSTAR DENDRIMERS A. Karbasou, Al Reza Ashraf Isttute of Naoscece ad Naotechology, Uversty of Kasha, Kasha , Ira ashraf@kashau.ac.r The Weer ad detour dces of a molecular graph G are defed as the sum of the legths of all shortest ad logest paths betwee the vertces of G. I ths paper the exact formulae for the Weer ad detour dces of a ew type of aostar dedrmers are gve. Key words: aostar dedrmer molecular graph Weer dex, detour dex ВИНЕРОВИ И ПОВРАТНИ ИНДЕКСИ ЗА НОВ ТИП ДЕНДРИМЕРИ ВО ФОРМА НА НАНОЅВЕЗДА За молекуларните графови се дефинирани Винерови (Weer) и повратни индекси кои се дефинирани како сума од должините на сите најкуси и најдолги патишта помеѓу пресеците во графот G. Во овој труд се изведени формули за Винеровите и повратните индекси за нов тип дендримери во форма на наноѕвезда. Kлучни зборови: dedrmer во форма на aoyвезда moleкularен graф Вerови deкси, povrate deкс INTRODUCTION Naostar dedrmers are part of a ew group of macromolecules that appear to be photo fuels ust lke artfcal ateas. The topologcal study of these macromolecules s the subect of ths artcle. Throughout ths paper a graph meas fte smple graph wthout multple edges ad loops. The set of vertces ad edges of a graph G are deoted by V(G) ad E(G), respectvely. The dstace d G (u,v) (d(u,v) for short) betwee two vertces u, v V(G) of a coected graph G s the legth of a shortest path coectg them. Suppose I deotes the set of all fte graphs ad R s the set of real umbers. A map Top from I to R s called a topologcal dex, f Top(H) = Top(G), for all pars (H,G) of somorphc graphs. The cocept of topologcal dex was frst proposed by Hosoya [] for characterzg the topologcal ature of a graph. Such graph varats are usually related to the dstace fucto d(-, -) : V(G) V(G) R. Recetly, ths part of Mathematcal Chemstry was amed "Metrc Graph Theory". The frst topologcal dex of ths type was proposed 947 by the chemst Harold Weer, []. It s defed as the sum of all dstaces betwee vertces of the graph uder cosderato. I the last meetg of Iteratoal Academy of Mathematcal Chemstry, professor Roberto Todesch aouced that he ad hs team mproved MOLE db Molecular Descrptors Data Base [, 4] for workg wth more tha thousad molecular descrptors. The MOLE db s a free o-le database costtuted of 4 molecular descrptors calculated o 477 molecules that allows the user to search for a specfc group of molecules ad aalyze the correspodg values of molecular descrptors ad to save a output fle wth the values of a block of

2 50 A. Karbasou, A. R. Ashraf molecular descrptors calculated o a group of molecules. Suppose G s a graph wth the vertex set V(G) = {v, v,, v }. I Metrc Graph Theory, the dstace matrx of G s defed as D(G) = [d ], where d = d(v,v ). The detour matrx DD = [dd ] ca be defed for G wth etres dd = 0 ad dd,, as the maxmum dstace betwee vertces v ad v. The detour matrx was troduced graph theory some tme ago by F. Harary [5] for descrbg the coectvty drected graphs. The detour matrx, cotrast to the dstace matrx that records the legth of the shortest path betwee vertces, records the legth of the logest dstace betwee each par of vertces. Ths matrx remaed ukow to chemsts utl the publcato by Amć ad Trastć [6]. The detour dex was troduced a year later by late Istva Lukovts [7]. Amć ad Trastć dscussed the detour dex ther paper, ad gave t the ame of Weer-lke dex. The computato of the detour matrx was preseted a paper by Trastć, Nkolć ad Mhalć [8] ad the whole story about the detour matrx, detour dex, ad ts uses chemstry are summarzed [9]. We ecourage the readers to cosult the book of Jaežč, Mlčevć, Nkolć ad Trastć [0] ad a paper by Joh [] ad the refereces there for more formato about ths topc. The problem of computg the topologcal dces of aostructures was rased by Dudea ad hs co-authors. I some research papers [ 8] they computed the Weer dex of aotubes ad tor. I [9, 0], the authors preseted some methods for calculato of the Weer dex ad resoace eergy of bezeod systems whch are extedable to aomaterals. I recet years, some authors worked o computg the Weer, PI, Schultz ad Szeged dces of the chemcal graphs of some aomaterals, [ 0]. Ths paper addresses the problem of computg the Weer ad detour dces of a fte class of aostar dedrmers. We choose these two topologcal dces because of ther correlatos wth some physco-chemcal propertes of molecules. Our otato s stadard ad take maly from the stadard books of graph theory. RESULT AND DISCUSSION Throughout ths paper G[] deotes the molecular graph of a aostar dedrmer wth exactly geerato (Fgs ). We frst calculate the Weer ad detour matrces of the graph G[] ad the compute the Weer ad detour dces of these aostars. At frst, we troduce two cocepts whch are mportat our calculatos. Suppose G ad H are graphs such that V(H) V(G) ad E(H) E(G). The we call H to be a subgraph of G. H s called sometrc, f for each x, y V(H), d H (x, y) = d G (x, y). I Fgure, four sometrc subgraphs of G[] are depcted. From ths fgure, t s clear that G[] s costructed from the subgraphs somorphc to B ad the core (Fg. ). To compute the Weer ad detour dces of G[], we calculate matrces WA, WA, WA ad WB whch are the Weer matrces of the subgraphs A, A, A ad B, respectvely. Suppose D ad D ' are 8 8 ad 8 60 matrces whch each etry s equal to ad M s the Weer matrx of the core. Fg.. The molecular graph of G[] Fg.. The core of G[] To costruct the Weer matrx of G[], t s eough to calculate the dstace matrx betwee a subgraph somorphc to B ad core, dstace matrx betwee two subgraphs somorphc to B (see A ad A Fg. ) ad the Weer matrx of the core. The dstace matrx betwee a subgraph somorphc to B ad core s equal to the sum of the Weer matrx of the subgraph A, WA, ad the matrx D ', where = l(p) such that P s a mmum path coectg a vertex of core to a vertex of B ad l(p) deotes the legth of P. We ow calculate the dstace matrx betwee two subgraphs somorphc to B. To do ths, we assume that B ad B are two subgraphs somorphc to B ad P s a mmum path coectg a vertex of B to a vertex of B. Obvously, there are two separate cases that oe of the ed vertces of P s a vertex of a hexago of G[] or two ed vertces of P are ot belog to a hexago. I the frst case, the dstace matrx D(B,B ) betwee B ad B s equal to WA + D ad for the secod D(B,B ) = WA + D. From Fg., oe ca see that to partto the Maced. J. Chem. Chem. Eg., 8 (), (009)

3 Weer ad detour dces of a ew type of aostar dedrmers 5 molecular graph of G[] to a core together wth four somorphc subgraphs M [],, M 4 []. We ame each of M [],, M 4 [], to be a brach of G ad M[] = M [] M 4 [].. Obvously, each of braches M [], 4, has exactly two somorphc compoets [ M ] ad M [ ]. Moreover, the core ad braches costtute a partto for G[]. Every subgraph M [], 4, has exactly + subgraphs somorphc to B such that degree of vertces of ther hexagos are G, say k,, l =, ad k =,,, 4. We ow defe s,, s 8 as follows: s s the summato of dstaces betwee vertces of ad as well as, ad 4, 4, for each of ad,, s s the summato of dstaces betwee vertces of,, ad,, ad 4,, 4, ad, 4, 4, ad, for each of ad,, s s the summato of dstaces betwee vertces of k ad k, for each of, ad k, ad k =,,,4, s 4 s the summato of dstaces betwee the vertces of [ M ] ad M [ ], s 5 s the summato of dstaces betwee vertces of k ad M k [ ], k s 6 s the summato of dstaces betwee vertces of,, from M [ ] ad 4, 4, from M [ ], s 7 s the summato of dstaces betwee vertces of M [] ad M [], as well as M [] ad M 4 [], s 8 s the summato of dstaces betwee other vertces of [ M ] ad M []. A A A Fg.. Some subgraphs of G[] I Table, the Weer matrx of G[] s computed. By defto of s,, s 8, oe ca prove the followg equaltes: B Table The Weer Matrx of G[]. C B B B B 4 B 5 B 6 B 7 B 8 C M A A A A A A A A B A B A +D A +D A +D 8 A +D 8 A +D 8 A +D 8 A +D B A A +D B A +D A +D A +D 8 A +D 8 A +D 8 A +D 8 B A A +D 8 A +D 8 B A +D 8 A +D 8 A +D A +D A +D B 4 A A +D 8 A +D 8 A +D B A +D 8 A +D 8 A +D A +D B 5 A A +D 8 A +D 8 A +D A +D B A +D A +D 8 A +D 8 B 6 A A +D 8 A +D 8 A +D A +D A +D B A +D 8 A +D 8 B 7 A A +D A +D A +D 8 A +D 8 A +D 8 A +D 8 B A +D B 8 A A +D A +D A +D 8 A +D 8 A +D 8 A +D 8 A +D B Maced. J. Chem. Chem. Eg., 8 (), (009)

4 5 A. Karbasou, A. R. Ashraf s =.(0+ ) = = s =.(0+ 8) = = s = 4..(0 7) = = = 9 9 l l k l s4 =.(5(l + ) (5k + )) = l= k= = s5 =.(5( + ) 7) = = = s6 =.(5.( + ) + 8) = = = s7 =..(5) =.(5( + ) + ) = = = = = l l k l+ + s8 =..(5(l + ) (5k + )) =.(5) = l= k= = = = By a smple calculato wth Maple, oe ca see that s + s s8 = Therefore we prove the followg theorem, Theorem : The Weer dex of G = G[] s computed as follows: ( ) = W G Proof. By defto of A, A, A, B, M, D ad D ' ad above calculatos, we have: W(G) = 64(s + s s ) + ( ) d + W(A )( 8) ' + 8, + W(A )( ) + W(A )( ) W(B)( 8) W(M) = ( ) 480( ) ( 8) + 46( ) + 68( ) ( 8) 558 = To compute the detour dex of G[], we defe the quattes t,, t 8 smlar to s,, s 8 by chagg dstace to logest dstace. Defe t,, t 8 as follows: t s the summato of maxmum dstaces betwee vertces of, as well as,, ad ad, 4, 4,for each of ad, -, t s the summato of maxmum dstaces betwee vertces of, ad,, ad 4,, 4 ad, 4, 4 ad,,for each of ad,, t s the summato of maxmum dstaces betwee vertces of k ad k, for each of, ad k, ad k =,,,4, t 4 s the summato of maxmum dstaces betwee the vertces of M[] ad M[ ], t 5 s the summato of maxmum dstaces betwee vertces of ad M k [ ], k k Maced. J. Chem. Chem. Eg., 8 (), (009)

5 Weer ad detour dces of a ew type of aostar dedrmers 5 t 6 s the summato of maxmum dstaces t 8 s the summato of maxmum dstaces betwee vertces of, from M [ betwee other vertces of M[] ad ] ad 4, 4 from M [ ], M[]. t 7 s the summato of maxmum dstaces By a smlar method as the oe above, t ca betwee vertces of M [] ad M [], as be see that the followg equatos are satsfed: well as M [] ad M 4 [], + B= 8 + A = 8 A = A = D = t =.(4 + 5) = = t =.(4 + ) = = t = 4..(4 9) = = = 9 9 l l k l t 4 =.(7(l + ) (7k + )) = l= k= = t5 =.(7( + ) 9) = = = t 6 =.(7.( + ) + ) = = = t 7 =.(7( + ) + 5) = = = t 8 +.(7) = = = = + A smple calculato by Maple shows that t + t t8 = Therefore we have the followg theorem: Theorem : The detour dex of G = G[] s computed as follows: ( ) = dd G Proof. By defto of A, A, A, B, M, D ad D ' ad above calculatos, we have: dd(g) = 64( ) + 480( ) + + 4( 8) + 608( ) + 59( ) ( 8) = Maced. J. Chem. Chem. Eg., 8 (), (009)

6 CONCLUSION I ths paper a ovel method for computg the Weer ad detour matrces of chemcal graphs are preseted. If a molecular graph G ca be decomposed to cycles ad paths the a smlar method as gve the paper ca be appled to compute the Weer ad detour matrces of G. So, the method gve ths paper s geeral for such molecular graphs. Ackowledgmet. The authors are greatly debted to the referees for ther valuable suggestos ad gvg hstorcal otes gve the troducto of ths paper. REFERENCES [] H. Hosoya, Topologcal Idex. A Newly Proposed Quatty Characterzg the Topologcal Nature of Structural Isomers of Saturated Hydrocarbos, Bull. Chem. Soc. Japa, 44, 9 (97). [] H. Weer, Structural Determato of Paraff Bolg Pots, J. Am. Chem. Soc., 69, 7 0 (947). [] MOLE db Molecular Descrptors Data Base, Mlao Chemometrcs ad QSAR Research Group, [4] R. Todesch ad V. Coso, Hadbook of Molecular Descrptors, WILE VCH, Wehem, 000. [5] F. Harary, Graph Theory, Addso-Wesley, Readg, Massachusetts, 969. [6] D. Amć, N. Trastć, O the Detour Matrx, Croat. Chem. Acta, 68, 5 6 (995). [7] I. Lukovts, The Detour Idex, Croat. Chem. Acta, 69, (996). [8] N. Trastć, S. Nkolć, Z. Mhalć, O Computg the Molecular Detour Matrx, It. J. Quatum Chem., 65, (997). [9] N. Trastć, S. Nkolć, B. Lučć, D. Amć ad Z. Mhalć, The Detour Matrx Chemstry, J: Chem. If. Comput. Sc., 7, 6 68 (997). [0] D. Jaežč, A. Mlčevć, S. Nkolć ad N. Trastć, Graph-Theoretcal Matrces Chemstry, Uversty of Kraguevac, Kraguevac, 007. [] P. E. Joh, Ueber de Berechug des Weer-Idex fuer ausgewaehlte Delta-dmesoale Gtterstrukture, MATCH Comm. Math. Comp. Chem.,, 07 9 (995). [] M. V. Dudea, A. Graovac, Geerato ad graphtheoretcal propertes of C 4 -tor, MATCH Commu. Math. Comput. Chem., 44, 9 0 (00). [] M. V. Dudea, I. Slagh-Dumtrescu, B. Parv, Toraes versus torees, MATCH Commu. Math. Comput. Chem., 44, 7 (00). [4] M.V. Dudea, P.E. Joh, Coverg polyhedral tor, MATCH Commu. Math. Comput. Chem., 44, 0 6 (00). [5] M. V. Dudea, Torodal Graphees from 4-Valet Tor, Bull. Chem. Soc. Jp., 75, (00). [6] M. V. Dudea, Hosoya Polyomal Tor, MATCH Commu. Math. Comput. Chem., 45, 09 (00). [7] P. E. Joh, M. V. Dudea, Weer dex of zg-zag polyhex aotubes, Croat. Chem. Acta, 77, 7 (004). [8] M.V. Dudea, M. Stefu, B. Parv, P. E. Joh, Weer dex of armchar polyhex aotubes, Croat. Chem. Acta,, 77, 5 (004). [9] D. Vukčevć, N. Trastć, Weer dces of bezeod graphs, Bull. Chemst & Techolgst Macedoa,, 9 (004). [0] Iva Gutma, Slavko Radekovc, A smple formula for calculatg resoace eergy of bezeod hydrocarbos, Bullet of the Chemsts ad Techologsts of Macedoa, 5, 7 (006). [] S. ousef, A. R. Ashraf, A exact expresso for the Weer dex of a polyhex aotorus, MATCH Commu. Math. Comput. Chem., 56, (006). [] S. ousef, A. R. Ashraf, A algorthm for costructg Weer matrx of TUC 4 C 8 (R) aotubes, Curret Naoscece, 4, 6 65 (008). [] A. R. Ashraf, S. ousef, Computg the Weer Idex of a TUC4C8(S) Naotorus, MATCH Commu Math Comput Chem, 57, (007). [4] S. ousef, A. R. Ashraf, A Exact Expresso for the Weer Idex of a TUC 4 C 8 (R) Naotorus, J Math Chem, 4, 0 09 (007). [5] A. R. Ashraf, S. ousef, A New Algorthm for Computg Dstace Matrx ad Weer Idex of Zg-zag Polyhex Naotubes Naoscale Res Lett,, 0 06 (007). [6] L. Xu, H. Deg, The Schultz Molecular Topologcal Idex of C 4 C 8 Naotubes, MATCH Commu. Math. Comput. Chem., 59, 4 48 (008). [7] S. Che, Q. Jag,. Hou, The Weer ad Schultz Idex of Naotubes Covered by C 4, MATCH Commu. Math. Comput. Chem., 59, (008). [8] M. Elas, B. Taer, Szeged Idex of Armchar Polyhex Naotubes, MATCH Commu. Math. Comput. Chem., 59, (008). [9] H. ousef-azar, A. R. Ashraf, A. Bahram, J. azda, Computg Topologcal Idces of Some Types of Bezeod Systems ad Naostars, Asa J. Chem., 0, 5-0 (008). [0] A. R. Ashraf, M. Mrzargar, PI, Szeged ad edge Szeged dces of a fte famly of aostar dedrmers, Ida Joural of Chemstry, 47A, (008).