The Reduced Overall Wiener Index

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1 BioChm Prss Intrnt lctronic Journal of Molcular Dsign 003,, Th Rducd Ovrall Winr Indx Xinhua Li,* Maolin Hu, Hongping Xiao School of Chmistry and Matrials Scinc, Wnzhou Normal Collg, Wnzhou, Zhjiang 3507, P.R.China Rcivd xxx; Prprint publishd xxx Accptd xxx ; Publishd xxx Intrnt lctron. J. Mol. Ds. 003,, Abstract Motivation. Comparison with th ovrall Winr indx, a novl st of ovrall Winr indxs ( m RW, m RW p, m RW pc, and m RW c ) wr dfind, which namd th rducd ovrall Winr indxs. Th potntial usfulnss of th rducd ovrall Winr indxs in QSAR/QSPR is valuatd by its corrlation with a numbr of C3-C8 alans and by a favorabl comparison with modls basd on th ovrall Winr indx and molcular connctivity indxs. This can xtnd th usfulnss of th Winr numbr, and can ma th Winr numbr to b a ind of widly usd topological indx in practic. Mthod. Th prsnt communication attmpts to corrlat boiling point, surfac tnsion, critical Tmpratur and critical prssurs of 38 divrs functional acyclic compounds with th rducd ovrall Winr indics and to compar thos with rlations of th ovrall Winr indx and molcular connctivity indics to xplor th diagnostic Rsults. Th rsults show that considrably bttr statistics is obtaind whn using th rducd ovrall Winr indx. Th rducd ovrall Winr indxs providd vn th sam statistical rsults as th molcular connctivity indxs in all modls with fiv variabls, and th standard dviations providd by ths two sts of indxs ar rathr closd. Conclusions. Summarizd th rsults, on may conclud that th rducd ovrall Winr indx xamind shows a good potntial for QSAR and QSPR studis. It dnots som information of molcular structur, and ma up for th pitfalls of Winr indx. This can xtnd th usfulnss of th Winr numbr, and can ma th Winr numbr to b a ind of widly usd topological indx in practic. Kywords. Winr indx; rducd ovrall Winr indx; molcular connctivity indx; Quantitativ structur proprty rlationships; QSPR; topological indx; molcular graph. Abbrviations and notations MLR, multipl linar rgrssion QSAR, quantitativ structur-activity rlationships QSPR, quantitativ structur-proprty rlationships m RW t, notations of th rducd ovrall Winr indxs INTRODUCTION Th Winr numbr W(G), on of widly usd dscriptors of molcular topology, was introducd in 947 by Winr[] as th path numbr for saturatd acyclic hydrocarbons. Th path numbr was dfind as th numbr of th bonds btwn all pairs of non-hydrogn atoms in th molcul[]. Th Winr numbr W(G) =W of a structur G is qual to th half-sum of distancs of th distanc matrix btwn all pairs of vrtics[3-4]: W ( G) = W = d () ij i j * Corrspondnc author; phon: ; fax: ; -mail: lixinhua0@63.nt This wor was supportd by Wnzhou scinc and tchnology projct of China (No. S003A008) and 55 Distinguishd Prson Foundation of Wnzhou. Copyright 003 BioChm Prss

2 Intrnt lctronic Journal of Molcular Dsign 003,, Rcntly, Bonchv proposs a nw approach to th topological charactrization of molculs[5-7]. It procds from th total numbr of subgraphs K(G) of molcular graph G and thir prsntation in complt sris of classs of ordr, with =0,,,, bing th constant numbr of dgs in ach subgraph of a crtain class, and bing th numbr of dgs in th ntir graph. H us th Winr numbrs of ach subgraph, xtnding thus that sminal molcular dscriptor to its most complt vrsion, th ovrall Winr indx[8]. In this study, according to this approach w dfind a rducd ovrall Winr indx. MATRIALS AND MTHODS According to th dfinition of Bonchv[8], th ovrall Winr indxs ar dfind as follows: Dfinition. Th ovrall Winr indx OW(G) of any graph G is dfind as th sum of th Winr indics W i (G i ) of all K subgraphs of G: K () OW G = W G G ( ) ( ) i= i Dfinition. Th th-ordr ovrall Winr indx OW(G) of any graph G is dfind as th sum of th Winr indics W j ( G j ) of all K subgraphs G j G that hav dgs: OW K ( G) = W j ( G j G) j= Dfinition 3. Th th-ordr ovrall Winr indx OW(G) can b prsntd as a sum of trms, OW (G), rprsnting th sum of th Winr indics in th subgraphs of spcifid typ. For acyclic graphs ths ar th path (=p), clustr(=c), and pathclustr (=pc) typ, as dfind by Kir and Hall[9]: K p K p K p (4) OW G OW G + OW G + OW G = W + W + W ( ) ( ) ( ) ( ) = p c pc pj cl j= l= m= Dfinition 4. Th ovrall Winr indx vctor OW (G) of any graph G is th squnc of all OW(G)s listd in an ascnding ordr of th numbr of dgs : ' 0 OW ( G) = OW{ OW, OW, OW, L, OW} (5) or in mor dtail for acyclic graphs: ' OW G = OW OW, OW, OW, OW, OW, L, OW, OW, OW (6) ( ) { } Whn w gt th ovrall Winr indxs OW RW and RW, thy ar dfid as: RW RW and = p i c p pcm c pc (3) OW, w can dfin th rducd ovrall Winr indxs OW N OW N Whr N is th numbr of atoms (xcpt hydrogn) of molculs. xampl: = (7) (8) BioChm Prss

3 Intrnt lctronic Journal of Molcular Dsign 00,, =: -, -3, 3-4, -5; RW = (4 )/5 = 0.6 =: --3, -3-4, --5, 5--3; RW = (4 4)/5 = 0.64 =3: --3-4, , --3-5; 3 RW = ( 0 + 9)/5 =.6 =3: th ntir graph; 4 RW = 8/5 = 0.7 RW = =.68; RW =.68 (0.6, 0.64,.6, 0.7) Tabl. Rducd ovrall Winr indx RW, and its th-ordr componnts, calculatd for C3-C8 alans no Molcul RW RW 3 RW 4 RW 5 RW 6 RW 7 RW RW propan n-butan mthyl propan n- pntan mthyl butan ,-dimthyl propan n-hxan mthyl pntan mthyl pntan ,3-dimthyl butan ,-dimthyl butan n- hptan mthyl hxan mthyl hxan thyl pntan ,4-dimthyl pntan ,3-dimthyl pntan ,-dimthyl pntan ,3-dimthyl pntan ,,3-trimthyl butan n-octan mthyl hptan mthyl hptan mthyl hptan thylhxan ,5-dimthyl hxan ,4-dimthyl hxan ,-dimthyl xan ,3-dimthyl hxan ,4-dimthyl hxan mthyl-3-thyl pntan ,3-dimthyl hxan mthyl-3-thyl pntan ,,4-trimthyl pntan ,3,4-trimthyl pntan ,,3-trimthyl pntan ,3,3-trimthyl pntan ,,3,3-ttramthyl butan Tabl. Path (=p), clustr(c), and path clustr (pc) trms of rducd ovrall Winr indx of C3-C8 alans no. Molcul 3 RW p 3 RW c 4 RW p 4 RW c 4 RW pc 5 RW p 5 RW c 5 RW pc 6 RW p 6 RW c 6 RW pc 7 RW p 7 RW c 7 RW pc BioChm Prss

4 Intrnt lctronic Journal of Molcular Dsign 003,, propan n-butan mthyl propan n- pntan mthyl butan ,-dimthyl propan n-hxan mthyl pntan mthyl pntan ,3-dimthyl butan ,-dimthyl butan n- hptan mthyl hxan mthyl hxan thyl pntan ,4-dimthyl pntan ,3-dimthyl pntan ,-dimthyl pntan ,3-dimthyl pntan ,,3-trimthyl butan n-octan mthyl hptan mthyl hptan mthyl hptan thylhxan ,5-dimthyl hxan ,4-dimthyl hxan ,-dimthyl xan ,3-dimthyl hxan ,4-dimthyl hxan mthyl-3-thyl pntan ,3-dimthyl hxan mthyl-3-thyl pntan ,,4-trimthyl pntan ,3,4-trimthyl pntan ,,3-trimthyl pntan ,3,3-trimthyl pntan ,,3,3-ttramthyl butan RSULTS AND DISCUSSION Winr numbr has two pitfalls, on is it shows opposing complxity trnds whn usd for a mixd st of isomric and nonisomric compounds, th othr is its dgnracy[8]. Ths two pitfalls prvnt th us of this numbr as a masur of molcular complxity. As th ovrall Winr indx prsrvs th basic faturs of th original Winr numbr, and liminating its pitfalls[8], our rducd ovrall Winr numbr can also liminating th pitfalls of Winr numbr, and can b a good masur of molcular complxity. Th potntial usfulnss of th rducd ovrall Winr indx for structur-proprty and structuractivity studis can b valuatd by prforming comparativ modling of rfrnc [8] against ovrall Winr indx and molcular connctivity indx, th most widly usd molcular dscriptors in QSAR/QSPR. In this study, w apply multipl linar rgrssion analysis (MLR) in a QSPR BioChm Prss 3

5 Intrnt lctronic Journal of Molcular Dsign 00,, modling of 4 physical proprtis of (C 3 -C 8 ) alans. Th proprtis ar boiling point, bp in ºC; hats of vaporization, HV in KJ/mol; critical prssurs, PC in atm; and surfac tnsion, ST in dyn/cm (Tabl 3.)[0]. Som of th modls of molcular connctivity ar tan from Bonchv[8], th bst modls of th rducd ovrall Winr indx hav bn slctd from th computr program. Th rducd ovrall Winr indics of th 38 molculs ar shown in Tabl and Tabl, and th physical proprtis of thirs ar shown in Tabl 3. Tabl 4 summarizs th statistics (corrlation cofficint r, standard dviation s, and th Fischr ratio F) of th molcular connctivity, ovrall Winr and rducd ovrall Winr indx modls of th C 3 -C 8 alans. Tabl 3. xprimntal valus for th physical proprtis of th 38 alans no. Molcul bp HV PC ST propan n-butan mthyl propan n- pntan mthyl butan dimthyl propan n-hxan mthyl pntan mthyl pntan dimthyl butan dimthyl butan n- hptan mthyl hxan mthyl hxan thyl pntan dimthyl pntan dimthyl pntan dimthyl pntan dimthyl pntan trimthyl butan n-octan mthyl hptan mthyl hptan mthyl hptan thylhxan dimthyl hxan dimthyl hxan dimthyl xan dimthyl hxan dimthyl hxan mthyl-3-thyl pntan dimthyl hxan mthyl-3-thyl pntan trimthyl pntan trimthyl pntan trimthyl pntan trimthyl pntan ttramthyl butan Th comparison of th thr columns of Tabl 4 shows that considrably bttr statistics rsults ar obtaind by th rducd ovrall Winr indx modls. Th rducd ovrall Winr indxs BioChm Prss 4

6 Intrnt lctronic Journal of Molcular Dsign 003,, providd vn th sam statistics rsults as th molcular connctivity indxs in all modls with fiv variabls, and th standard dviations providd by ths two sts of indxs ar rathr clos. Tabl 4. Comparison of multi linar rgrssion statistics (corrlation cofficint, standard dviation, and th Fischr ratio) for th C3-C8 alan proprtis modls with 5 paramtrs producd by thr sts of topological indxs. Numbr of indx r s F Boiling Point Ovrall Winr indx Molcular connctivity indx Rducd ovrall Winr indx HV Ovrall Winr indx Molcular connctivity indx Rducd ovrall Winr indx Critical Prssur Ovrall Winr indx Molcular connctivity indx Rducd ovrall Winr indx Surfac Tnsion Ovrall Winr indx Molcular connctivity indx Rducd ovrall Winr indx Th bst modls of rducd ovrall Winr with fiv variabls obtaind for th 4 xamind alan proprtis ar giv blow. Th first ordr rducd ovrall Winr indx tas part in all modls. bp = RW RW RW p RW RW pc (9) n = 38, r = , s =.57, F = 903 HV = RW RW RW p RW RW p (0) n = 34, r = 0. 99, s = 0.46, F = 75 PC = RW RW RW pc RW RW c () n = 38, r = , s = 0.30, F = 3 ST = RW RW RW RW p RW p () n = 33, r = , s = 0.6, F = 75 4 CONCLUSIONS Summarizd th rsults, on may conclud that th rducd ovrall Winr indxs xamind show a good potntial for QSAR and QSPR studis. Th rsults show that considrably bttr statistics is obtaind whn using rducd ovrall Winr indxs. Th rducd ovrall Winr indxs providd vn th sam statistics rsults as th molcular connctivity indxs in all modls with fiv variabls, and th standard dviations providd by ths two sts of indxs ar rathr closd. Th rducd ovrall Winr indx can b a good masur of molcular complxity. This can xtnd th usfulnss of th Winr numbr, and can ma th Winr numbr to b a ind of BioChm Prss 5

7 Intrnt lctronic Journal of Molcular Dsign 00,, widly usd topological indx in practic. 5 RFRNCS [] H. Winr, Structural Dtrmination of Paraffin Boiling Points, J.Am.Chm.Soc. 947, 69, 7-0. [] J. R. Platt, Influnc of Nighbor Bonds on Additiv Bond Proprtis in Paraffins, J. Chm. Phys. 947, 5, [3] H. Hosoya, Topological indx: A nwly proposd quantity charactrizing th topological natur of structural isomrs of saturatd hydrocarbons, Bull. Chm. Soc. Jpn. 97, 44, [4] J. Plsni, On th sum of all distancs in a graph or digraph, J.Graph. Thory. 984, 8, -. [5] D. Bonchv, Kolmogorov s Information, Shannon s ntropy, and Topological Complxity of Molculs, Bulg.Chm.Commun. 995, 8, [6] D. Bonchv, Novl Indics for th Topological Complxity of Molculs, ASR/QSAR nvir.rs. 997, 7, [7] D. Bonchv, Ovrall Connctivity/Topological Complxitis: A Nw Tool for QSAR/QSPR. J.Chm.Inf.Comput. Sci. 000, 40, [8] D. Bonchv, Th Ovrall Winr Indx-A Nw Tool for Charactrization of Molcular Topology, J.Chm.Inf. Comput.Sci, 00, 4, [9] L.B. Kir, L.H. Hall, Molcular Connctivity in Chmistry and Drug Rsarch. Acadmic Prss: Nw Yor 976. [0].N.Dian, I.C. Wi, and P.G. Sybold, Molcular modling of th physical proprtis of alans, J.Am.Chm. Soc. 988, 0, Biographis Xinhua Li is associat profssor of physical chmistry at th Wnzhou normal collg. Aftr obtaining a Ph.D. dgr in physical chmistry from th Zhjiang Univrsity, Dr. Xinhua Li undrtoo postdoctoral rsarch with Profssor Yuchun Hou at th Zhjiang Univrsity. Mor rcntly, Dr. Xinhua Li has collaboratd on projcts with Profssor duardo A. Castro. Maolin Hu is associat profssor of inorganic chmistry at th Wnzhou normal collg. H obtaind a Mastr dgr in inorganic chmistry from th Zhjiang Univrsity. Hongping Xiao is assistant profssor of inorganic chmistry at th Wnzhou normal collg. H obtaind a Mastr dgr in inorganic chmistry from th Zhjiang Univrsity. BioChm Prss 6