Advances of Clar's Aromatic Sextet Theory and Randic 's Conjugated Circuit Model

Transcription

1 Th Op Orgac hmstry Joural 0 5 (Suppl -M6) Op Accss Advacs of lar's Aromatc Sxtt Thory ad Radc 's ougatd rcut Modl Fu Zhag a Xaofg Guo a ad Hpg Zhag b a School of Mathmatcal Sccs Xam Uvrsty Xam Fua 05 ha b School of Mathmatcs ad Statstcs Lazhou Uvrsty Lazhou Gasu ha Abstract: lar's aromatc sxtt thory provds a good mas to dscrb th aromatcty of bzod hydrocarbos whch was maly basd o xprmtal obsrvatos lar dfd sxtt pattr ad lar umbr of bzod hydrocarbos ad h obsrvd that for somrc bzod hydrocarbos wh lar umbr crass th absorpto bads shft to shortr wavlgth ad th stablty of ths somrs also crass Motvatd by lar's aromatc sxtt thory thr typs of polyomals (sxtt polyomal lar polyomal ad lar covrg polyomal) wr dfd ad Radć's cougatd crcut modl was also stablshd I ths survy w attmpt to rvw som advacs o lar's aromatc sxtt thory ad Radc 's cougatd crcut modl th past two dcads Nw applcatos of ths polyomals to fullrs ad calculato mthods of lar dpdt ad mmal cougatd crcut polyomals of bzod hydrocarbos ar also prstd Kywords: Sxtt polyomal lar polyomal lar covrg polyomal Larly dpdt ad mmal cougatd crcut polyomal -Rsoac Bzod hydrocarbo Fullr INTRODUTION Th aromatcty rflcts xtra stablty of crta typs of cougatd systms du to th atur of molcular orbtals Th rsoac rgy calculatd from xprmtal masurmts dots th rgy ga or loss du to th tracto btw Kulé structurs ad rprsts th xtra stablty of th cougatd systm Thr hav b dstct approachs dvlopd to stmat th rsoac rgy To dal wth th problm smmprcal valcbod vw dffrt VB basd modls (s [] for dtals) wr bult succssvly ad hrarchcally followg Paulg ad Whlad [] Amog ths modls lar's aromatc sxtt thory s maly basd o xprmtal obsrvatos whch dscrbs th aromatcty of bzod hydrocarbos I ths survy w attmpt to rvw som advacs o lar's aromatc sxtt thory ad Radc 's cougatd crcut modl for polyhxs ( bzod hydrocarbos (bzod systms)) ad corood hydrocarbos (corood systms) as wll as fullrs W frst dscuss thr polyomals o th aromatcty of polyhxs Th w xtd th dscussos to vstgat th stablty of fullrs Th sourcs of lar's aromatc sxtt thory sm to b th papr of chmsts Armt ad Robso [] ad th wor of physcal chmst Hücl [4-6] Th mportat rol of 6- mmbrd cougatd cycls amog 4 cougatd cycls may also hav sprd lar's aromatc sxtt thory sxtt polyomal lar polyomal ad lar covr polyomal I lar's aromatc sxtt thory dlocalzd Addrss corrspodc to ths author at th School of Mathmatcal Sccs Xam Uvrsty Xam Fua 05 ha; Tl: ; Fax: ; E-mal: fzhag@xmuduc lctros a 6-cougatd cycl ar dotd by a crcl ad as statd Gutma [7] a lar structur (lar formula) cosstg of crcls satsfs th followg thr ruls: (a) rcls ar vr draw adact hxagos (b) Th rmadr of th polyhx obtad by th dlto of th vrtcs of th hxagos that possss crcls must b mpty or hav a Kulé structur ad (c) As may crcls as possbl ar draw subct to th costrats (a) ad (b) If w draw som crcls wth oly th costrats (a) ad (b) w obta a gralzd lar structur (or sxtt pattr) lar obsrvd that for somrc bzod hydrocarbos wh th umbr of th crcls of lar structurs (calld lar umbr) crass th absorpto bads shft to shortr wavlgth ad th stablty of th somrs also crass I hs boo [8] lar provdd may xampls to support hs obsrvato ad bult lar's aromatc sxtt thory Rctly topgraphcal faturs of th molcular lctrostatc pottal of a srs of polycyclc aromatc bzod hydrocarbos hav b aalyzd at BLYP/6-G(dp) ad MP/6- G(dp) lvls Th thortcal rsults fully support lar's aromatc sxtt thory [9] I th study of lar's aromatc sxtt thory th frst tas s to dtrm th lar umbr For small bzod hydrocarbos w ca asly fd a lar structur by p ad papr Th lar structurs of som larg bzod hydrocarbos ar xmplfd Fg () For ths xampls th codtos (a) ad (b) ca b asly vrfd whl th codto (c) o th maxmalty of th umbr of crcls was provd [0] For mor xampls ad furthr dscussos th radr s rfrrd to [-6] / 0 Btham Op

2 88 Th Op Orgac hmstry Joural 0 Volum 5 Zhag t al Fg () lar formulas of som classs of bzod systms possssg x aromatc sxtts Has ad Zhg [7 8] computd th lar umbr of a bzod hydrocarbo by th tgr lar programmg ad cocturd that th lar programmg rlaxato was suffct for th gral cas Th coctur has b provd by Abldo ad Atso [9] A furthr problm s whthr w ca rf th da of lar umbr A rasoabl assumpto s that all gralzd lar structurs cotrbut to th rsoac rgy From ths pot of vw Hosoya ad Yamaguch dfd th sxtt polyomal [0] Aothr rasoabl assumpto s that th maxmal gralzd lar structurs (whch ar ot a propr subst of aothr gralzd lar structur) cotrbut th most to th rsoac rgy From ths pot of vw El- Basl ad Rad c dfd th lar polyomal [] Isprd by lar structur w rasoably thought that for th rsoac rgy of a bzod hydrocarbo bsds th aromatc 6-crcuts all doubl bouds should also b cosdrd Ths motvatd us to df th lar covr of a bzod hydrocarbo ad th coutg polyomal lar covrg polyomal [] Th cougatd crcut modl a rsoac-thortc modl was troducd by Rad c [-5] 976 for th study of aromatcty ad cougato polycyclc cougatd systms It cosdrs cotrbutos of ot oly 6 -mmbrd rgs but also all (4 ) -mmbrd cougatd crcuts as wll as gatv cotrbutos of 4 - mmbrd cougatd crcuts to th rsoac rgy Th modl was motvatd from a mprcal pot of vw laboratg th lar aromatc sxtt thory [8] Th cougatd-crcut modl has also a frm quatum mchacal bass [ 6 7] It ca b drvd rgorously from th Paulg-Whlad rsoac thory [8-] va a Smpso-Hrdo modl Hamltoa [-4] Th cougatd crcut modl ca b appld to mor gral cass For xampl Maoharaa t al [5] vstgatd th stablty of fullrs prdctd by th topologcal rsoac rgy (TRE) modl ad th cougatd crcut modl ad Bab c ad Trast c [6] rportd th rsoac rgs (REs) of svral fullrs wth 4-mmbrd rgs ad thr somrs wth oly 5- ad 6-mmbrd rgs usg th cougatd-crcut modl ad th TRE modl Hr w cosdr oly th cougatd crcut modl for bzod hydrocarbos Th lar dpdt ad mmal cougatd crcut polyomals ( LM - polyomals) ar dfd xplctly ad a rcursv mthod ad aalytcal xprssos for calculato of LM - polyomals ar dscussd Fally w mto som rsults o -rsoac ( - cycl rsoac) polyhxs op d aotubs torodal polyhxs Kl-bottl polyhxs ad fullrs as wll SEXTET POLYNOMIAL Lt G dot a bzod systm A gralzd lar structur ( sxtt pattr) of G s a st of dsot hxagos of G ach hxago of whch a crcl s draw such that th dlto of th vrtcs of such hxagos togthr wth thr cdt dgs rsults a graph wth a prfct matchg or a mpty graph To cout sxtt pattrs of a bzod hydrocarbo G Hosoya ad Yamaguch [0] dfd th sxtt polyomal B ( x ) as follows ( G) BG ( x)= r( G ) x () =0 whr rg ( ) s th umbr of sxtt pattrs of G wth hxagos (or gralzd lar structurs wth cycls) ad G ( ) s lar umbr th maxmum sz of sxtt pattrs Th cocpts of sxtt pattr ad lar umbr hav b aturally xtdd to polycyclc cougatd hydrocarbos such as corood systms carbo aotubs ad fullrs tc Th sxtt polyomal has som trstg mathmatcal proprts Hosoya ad Yamaguch [0] ad Oham ad Hosoya [7] foud that thr s a o-to-o corrspodc btw th sxtt pattrs ad th Kulé structurs for a catacodsd bzod hydrocarbo G G

3 Advacs of lar's Aromatc Sxtt Thory Th Op Orgac hmstry Joural 0 Volum 5 89 BG () = K( G ) () whr KG ( ) dots th umbr of Kulé structurs of G Thy cocturd that for ay bzod hydrocarbo (polyhx graph) whch has at last o Kulé pattr thr xsts a o-to-o corrspodc btw Kulé structurs ad sxtt pattrs Two proofs of th coctur wr gv by H ad H [8] ad by Oham [9] But th proofs ar ot complt ad hav som rrors Zhag ad Guo [40 4] gav a xplct dfto of supr sxtts of gralzd polyhxs ad a w proof of th Oham- Hosoya coctur Zhag ad h [4] showd that ach hxago of a bzod systm B forms a sxtt pattr rb ( ) s qual to th umbr of hxagos of B f ad oly f B s ormal Th smlar rsults hold for ormal corood systms [4] ad pla lmtary bpartt graphs [44] Th sxtt polyomal ca b formally dffrtatd wth d rspct to x as B G( x)= BG( x) whr th drvatv ca dx b xprssd as th sum of sxtt polyomals of som subgraphs of G Thorm [45] Lt G b a bzod systm Th B ( x)= B ( x) () G h Gh whr th summato gos ovr all hxagos h of G ad G h dots th subgraph obtad from G by dltg hxago h wth cdt dgs ad vrtcs B G () Rad c [46] potd out that th quott ca b BG () rgardd as a masur of th total aromatcty of a bzod G I th followg subsctos w troduc som mathmatcal proprts ad chmcal applcatos of sxtt polyomals varous chmcal graphs Bzod has ad yclo-polyphacs How to comput th sxtt polyomal for a bzod hydrocarbo? Ths s a problm of mportac It s wll ow that thr ar rcurrc rlatos for computg almost all polyomals wth applcatos chmstry such as th charactrstc polyomal th dpdt polyomal ad th matchg polyomal of a graph I [47] Gutma t al gav a mthod of rcurrc to comput th sxtt polyomals for cata-codsd bzods Gutma [48] foud that for ay bzod cha (ubrachd catacodsd bzod hydrocarbo) thr s a bcto btw ts gralzd lar structurs ad -matchgs of th corrspodg Gutma (catrpllar) tr Rcall that graph thory a catrpllar tr s a tr whch th rmoval of all ts pdat vrtcs (vrtcs of dgr ) rsults a path I othr words lt v v b a path If w o ach of vrtcs to a vrtx v by a dg = th a catrpllar tr s obtad For a bzod cha B ts corrspodg Gutma tr s dfd by th followg costructo: For th hxagos ad th d hxagos of B rprst ach of thm by a dg Th o ths dgs succssvly to obta a path If B cotas (lar aulatd) hxagos btw th hxagos corrspodg to th succssv dgs v v ad vv th w add w vrtcs ad o ach of thm to v by a dg (s Fg for a xampl) Gutma [48] ad El-Basl [49] provd th followg rsult Fg () A bzod cha ad ts corrspodg Gutma tr Thorm [48 49] Lt B b a bzod cha ad G ts corrspodg Gutma tr Th rb ( )= mg ( ) whr rb ( ) s th umbr of sxtt pattrs of B wth hxagos ad mg ( ) s th umbr of -matchgs of G osdrg th cotrbuto of lar structur to th rsoac rgy w troduc a quas-ordr o bzod hydrocarbo somrs to compar thr rsoac rgy For two bzod somrs B ad B f rb ( ) rb ( ) for =0 th w say B s s -gratr tha B ad wrt B > B If both B > B ad B > B hold th B ad B ar sad to b s -quvalt If thr B > B or B < B holds th B ad B ar comparabl larly two s - quvalt bzod chas may hav th sam sxtt polyomal but d ot b somorphc Basd o th umbr of -matchgs of a graph w ca df a smlar quas-ordr ( m -gratr) for graphs wth th sam umbr of vrtcs If for two graphs G ad G mg ( ) mg ( ) = 0 th w say G s m - gratr tha G ad wrt G G [50] By Thorm w hav: Thorm [5] Lt B ad B b two bzod chas wth th sam umbr of hxagos ad G ad G ar th Gutma trs of B ad B rspctvly Th B > B f ad oly f G G Usg Thorms ad w ca dtrm th xtrm bzod chas wth rspct to thr lar aromatc

4 90 Th Op Orgac hmstry Joural 0 Volum 5 Zhag t al sxtts I [5] t s showd that th mmal bzod cha s a lar cha ad th last four mmal bzod chas ar also dtrmd O th othr had th maxmal bzod chas ar -cycl rsoat bzod chas (th bzod chas whos hxagos ar all s xcpt th frst ad last o) ad th scod os ar th bzod chas whos hxagos ar all s xcpt th frst ad last two Not that [5] cotas a rror dscrbg th scod maxmal bzod chas Smlar to [5] cocrg th ordrg for bzod chas th authors also cosdr th ordrg of cyclopolyphacs (cludg th spcal cas of prmtv coroods) Th cyclo-polyphacs ca b obtad from a cha of hxagos by dtfyg o dg of a d hxago wth a dg of th othr d hxago so that ach hxago s adact to xactly two hxagos W ca comput th sxtt polyomal of cyclo-polyphacs rcurrtly wth th hlp of matchg polyomal For somrc cyclo-polyphacs w troduc a quas-ordr to compar thr rsoac rgy For ths am w d to df th gralzd crow corrspodg to a cyclopolyphac Rcall that a gralzd crow s a graph whch th rmoval of all ts d vrtcs (vrtcs of dgr ) rsults a cycl I othr words lt vv vv b a cycl If w o ach of m w vrtcs by a dg to th vrtcs v for = th a gralzd crow s obtad For a cyclo-polyphac B w ca df a corrspodg gralzd crow (as llustratd Fg ) ad prov th followg thorm Thorm 4 [5] Lt B b a cyclo-polyphac ad G b ts corrspodg gralzd crow Th th umbr of sxtt pattrs of B havg prcsly hxagos s qual to th umbr of -matchgs of G for ay o-gatv tgr Thorm 4 was usd to solv ``Hosoya's mystry" [5 54] cocrg th cocdc btw th charactrstc polyomal of a cycl ad th polyomal of Kulé structur cout of a prm corood For dtals th radr s rfrrd to [55] Wth th hlp of matchg polyomal Thorm 4 ca b also usd to comput sxtt polyomals of cyclopolyphacs Som xampls wr gv [5] Smlar to th cas of bzod chas th ordrg of cyclopolyphacs ca also b st up Th w hav Thorm 5 [5] Lt B ad B b two cyclopolyphacs wth th sam umbr of hxagos wth G ad G thr corrspodg gralzd crows rspctvly Th G G ( strctly ) f ad oly f B > B ( strctly ) Thorm 4 rducs th ordrg problm of cyclopolyphacs (wth fxd umbr of lar structurs) to th ordrg problm of th umbr of -matchgs of gral crows Usg Thorms 4 ad 5 ad som old rsults [50] th authors of [5] dtrmd th mmal scod mmal to svth mmal cyclo-polyphacs wth rspct to th umbr of lar's sxtts Thy also dtrmd th maxmal ad scod maxmal famly of cyclopolyphacs wth rspct to thr umbr of lar structurs For dtals th radr s rfrrd to [5] Rsoat Pattrs ad Kulé Structurs - Altrat as Th o-to-o corrspodc btw th sxtt pattrs ad Kulé structurs was frst rvald for th bzods by Hosoya ad Yamaguch as follows Thorm 6 [0] For a catacodsd bzod systm H BH () = K( H ) For th coro G (s Fg 4) ts sxtt polyomal B ( )= 7 9 G x x x x So BG ()=9< K( G )=0 I fact th coro s th crtcal forbdd subgraph for th abov rlato () Ths ca b xprssd th followg thorm obtad by Zhag ad h [56] whch was rprovd latr a ovl approach [57] A subgraph H of a graph G s calld c f G V( H) thr has a prfct matchg or s mpty Thorm 7 [56] For a hxagoal systm H wth prfct matchgs BH () K( H) ad qualty holds f ad oly f H cotas o coro as ts c subgraph By troducg supr rgs sxtt pattrs of a bzod systm a gral o-to-o corrspodc btw sxtt pattrs ad Kulé pattrs ca b stablshd (s [40 4 For xampl th xtror boudary of coro as a supr rg s addd to th ctral hxago to produc a w sxtt pattr For th gral altrat cas -- pla bpartt graphs G wth prfct matchgs Gutma [58] ad Joh [59] Fg () A cyclo-polyphac ad ts corrspodg gralzd crow

5 Advacs of lar's Aromatc Sxtt Thory Th Op Orgac hmstry Joural 0 Volum 5 9 dpdtly dfd rsoat polyomals ad cll polyomals to b cout polyomals of rsoat pattrs Thy xtd sxtt polyomals of bzod systms by rplacg hxagos of a sxtt pattr wth v r facs Lt rg ( ) dot th umbr of rsoat pattrs of G Joh t al [] obtad Thorm 8; ad a rfd rsult -- Thorm 9 was obtad [6 6] Fg (5) oraul For th wll-ow fullr--cosahdro (Fg 6 (lft)) El-Basl [67] frst foud that ts lar umbr quals 8 Sc t has th Frs structur so that ach hxago s altratg ay st of dsot hxagos always forms a sxtt pattr Basd upo ths Shu t al [68] computd th sxtt polyomal of as Fg (4) oro Thorm 8 [] For a pla bpartt graph G rg ( ) KG ( ) Thorm 9 [6 Thorm ] Lt G b a - coctd pla bpartt graph wth prfct matchgs Th rg ( ) KG ( ) ad qualty holds f ad oly f thr do ot xst dsot cycls R ad such that (a) R s a facal boudary lyg th tror of ad (b) R s a c subgraph of G Rsoat Pattrs ad Kulé Structurs - No- Altrat as From th abov subscto w s that th corrspodc btw rsoat pattrs ad Kulé pattrs rls strogly o th xstc of th root prfct matchg of a pla bpartt graph [6 64] Ths s ot sutabl for th o-altrat cas (o-bpartt pla graphs) By applyg a ovl approach mathmatcal ducto ad th prcpl of cluso ad xcluso combatorcs Zhag ad H [65] showd that for ay pla graphs th umbr of prfct matchgs s ot lss tha th umbr of rsoat pattrs Ths gralzs th corrspodg rsults bzod systms ad pla bpartt graphs Applcatos to fullrs (plaar cubc graphs wth oly ptagoal ad hxagoal facs) wr also dscussd For xampl th rsoat pattrs of coraul (Fg 5) ar as follows: {}{ } { }{ 4}{5}{ }{4}{ 4}{ 5}{ 5} Hc B ( x)= 5x 5x coraul It s computd [66] that K (coraul) = That s r(coraul) = K (coraul) = B x x x x x x x x x ( ) = (8) Y t al [69] showd that vry hxago of a fullr s rsoat dtrmd all th othr ght -rsoat fullrs ( vry st of at most thr dsot hxagos forms a sxtt pattr) ad provd that ay dpdt hxagos of a -rsoat fullr graph form a sxtt pattr So th sxtt polyomals of th othr ght -rsoat fullr graphs ar computd by coutg sts of dsot hxagoal facs (s [69] for dtals) For such -rsoat fullr graphs F w ca cofrm that th BF () < K( F ) I gral Zhag ad H showd th followg rsult Thorm 0 [65] For ay pla graph G rg ( ) KG ( ) For all fullr graphs Sr ad Sthlí [70] provd th followg rsult whch was cocturd arlr by Zhag ad H [65] Thorm [70] For vry fullr graph F BF () < K( F ) 4 Stablty Idcators For bzod hydrocarbos both th lar umbr ad Kulé cout ca masur thr stablts Howvr Aust t al [7] costructd 0 dstct fullr somrs of whos Kulé couts surpass th Kulé cout (500) of cosahdral So th maxmalty of Kulé couts of fullr somrs may ot corrspod to th hghst stablty Zhag t al [7] turd to vstgatg a sgfcat rol of th lar umbrs of fullrs thr stablts Zhag ad Y [7] obtad th sharp uppr boud for th lar umbr of fullrs as follows Thorm [7] Lt F b a fullr wth vrtcs Th cf ( ) 6 Thy also showd that thr ar ft may fullr graphs whos lar umbr ca achv ths uppr boud

6 9 Th Op Orgac hmstry Joural 0 Volum 5 Zhag t al Fg (6) lar formulas of (:8) (lft) ad 70 (70:849) (rght) cludg ad 70 ad zgzag ad armchar carbo aotubs as wll; Thorm shows that o of fullr graphs ar ``all-bzods" [74 75] ombg thorm ad som costructo of lar formula of fullrs thy foud that th xprmtally charactrzd : D (IV) ad 84 : D d (II) atta th maxmum lar umbr amog thr fullr somrs Y ad Zhag [76] hav rgorously provd that xactly 8 fullrs wth atoms (cludg th cosahdral ) achv th maxmum lar umbr 8 A comparso shows that o of ths 8 fullrs blogs to th collcto of th 0 fullr somrs wth Kulé couts surpassg 500 [7] That s th lar umbrs of ths 0 fullr somrs ar all lss tha 8 Hc a combato of lar umbr ad Kulé cout as a stablty prdctor dstgushs uquly th cosahdral from ts all 8 fullr somrs Furthrmor W Su ad F Wag Lazhou Uvrsty hav computd th sxtt polyomals of all fullr somrs of ad 70 It s ow that ad 70 hav 8 ad 849 fullr somrs rspctvly [77] From thr computatoal rsults w ca s that (:8) s a uqu fullr somr of wth th maxmum umbr of sxtt pattrs 588 ad th somr :809 has th scod maxmum umbr of sxtt pattrs 970 So (:8) has a much largr sxtt pattr cout tha th somr :809 Smlarly th xprmtally charactrzd 70:849 s a uqu fullr somr of 70 wth th maxmum sxtt pattr cout 874 ad 70:706 has th scod maxmum sxtt pattr cout 746 Such partal computatoal rsults ar lstd Tabl How to comput lar umbrs of fullrs s a trstg problm both mathmatcs ad thortcal chmstry Up to ths dat a ffctv gral way has ot b foud for ths problm It s worthwhl to s approprat combatos of lar umbrs wth othr varats as stablty prdctors of fullrs LAR OVERING POLYNOMIALS Th dfto for a sxtt pattr of a gralzd bzod systm B was slghtly modfd [] W add a Kulé structur of H Q to a sxtt pattr Q to gt a vrtx-covr of B ad w call such a vrtx-covr a lar covr of a (gralzd) bzod systm B I othr words a spag subgraph of B s sad to b a lar covr of B f ach of ts compots s thr a hxago or a dg Th th lar covrg polyomal of B s dfd as: ) ( x)= x)= z ) x (4) =0 whr zb ( ) dots th umbr of lar covrs of B havg prcsly hxagos (rfr to [ 78-8 Ths polyomal was usd to covtly compar topologcal dcs of som typs of bzod somrs [80] It s also calld ``Zhag-Zhag polyomal" a srs of paprs du to Gutma t al [8-88] Tabl Th Frst Scod ad Thrd Maxmum B F () Fullr Isomrs of ad 70 Isomrs F Sxtt polyomal B F(x) B F() :804 x8 48x7 77x6 94x5 06x4 588x 57x 0x 87 : x 98x 594x 50x x 66x 58x 0x 970 : x 0x 40x 9x 50x 6x x 0x :776 x9 0x8 87x7 4478x6 545x5 6x4 07x 5x 5x : x 4x 90x 4500x 5474x 64x x 5x 5x : x 75x 065x 475x 5958x 940x 55x 55x 5x 874

7 Advacs of lar's Aromatc Sxtt Thory Th Op Orgac hmstry Joural 0 Volum 5 9 Basc Proprts W frst troduc som basc proprts of lar covrg polyomal of a gralzd bzod systm B Thorm [] Lt B b a bzod systm Th w hav th followg proprts for th lar covrg polyomal of B : 0) = K) th dgr of th polyomal x ) s B ( ) th lar umbr of B th coffct of th hghst dgr trm zbb ( ( )) quals th umbr of lar formulas of B 4 zb ( ) = h ) th frst Hrdo umbr Th lar covrg polyomal closly rlats to th sxtt polyomal va a trasformato of polyomals Fg (7) Mods of propr sxtt ad mpropr sxtt Lt B b a bzod systm wth a prfct matchg (Kulé structur or -factor) M A cougatd (or altratg) hxago of B s calld a propr sxtt f th xtrm rght vrtcal dg blogs to M ; a mpropr sxtt othrws as llustratd Fg (7) Us Lt ab ( ) dot th umbr of prfct matchgs of B whch cotas prcsly propr sxtts for 0 ) Th w hav ) ab ( )= KB ( ) ab ( )> 0 for all 0 ) ad =0 ab ( 0)= Furthr ths ab ( ) bcom th coffcts of a w polyomal xprsso for th lar covrg polyomal varabl ( x ) Thorm [79] Lt B b a bzod systm wth a prfct matchg Th th lar covrg polyomal x ) of B ca b xprssd th followg form: ) ) x )= zbx ( ) = ab ( )( x ) =0 =0 Va such a trasformato w ca stablsh a rlato wth th sxtt polyomal as follows Thorm [79] Lt B b a bzod systm wth a prfct matchg For all 0 ) ab ( ) rb ( ) ad all th qualts hold f ad oly f B has o coro (s Fg 4) as ts c subgraph orollary 4 [79] Lt B b a bzod systm wth a prfct matchg Th ) abx ( ) s th sxtt polyomal of B f ad oly f =0 B has o coro as ts c subgraph For othr trstg proprts th radr s rfrrd to [79] ) I gral th polyomal abx ( ) ca b vwd =0 as th rvsd sxtt polyomal of B whch couts sxtt pattrs wth supr rgs omputato Approach ompard wth th sxtt polyomal th lar covrg polyomal of a bzod systm has o advatag computato: t has gral rcurrc rlatos Ths abls o to comput som sgfcat topologcal dcs of bzod systms as mtod Thorm by som rcurrc procdurs Thorm 5 [] Lt B b a gralzd bzod systm wth th compots B B B Th x )= x) = Thorm 6 [] Lt B b a gralzd bzod systm Lt s ad s b two hxagos of B havg a commo dg = xy (s Fg 8 (lft)) Th )= w s ) xy) x y) = whr B s dots th subgraph obtad from B by dltg all vrtcs of s togthr wth cdt dgs Thorm 7 [] Lt B b a gralzd bzod systm Lt xy b a dg of a hxago s of B whch ls o th prphry of B (s Fg 8 (mddl)) Th )= w s) x y) xy) Fg (8) Mods of hxagos s ad s a bzod systm for som rducd procdurs

8 94 Th Op Orgac hmstry Joural 0 Volum 5 Zhag t al Thorm 8 [] Lt X ad X b two Kuléa bzod systms whch cota hxagos s ad s rspctvly as dcatd Fg 8 (rght) (or o of thm b K ) Lt X: X b a bzod systm obtad by glug X ad X oly alog a dg xy of s ad s Th th lar covrg polyomal of X: X s ( X : X )= ( X ) ( X ) ( X ) ( X ) ( X ) ( X ) whr X = X x y for = or orollary 4 ad Thorm 8 ca b usd to drv a rcurrc rlato for sxtt polyomal of cata-codsd bzod systms as follows orollary 9 [6 orollary 5] Lt X b a catacodsd bzod systm Th B ( x)= mxb ( x) B ( x) L m : X X X whr L m dots th lar bzod cha of m hxagos lar covrg polyomals hav b computd for of som typs of bzod ad corood systms such as bzod chas [] paralllogram [6] multpl lar hxagoal chas [88] cyclo-polyphacs [89] ad so o [90 9] Applcatos to Rsoac Ergy Each of th quatts mtod pots ()-(4) of Thorm was show to rlat to som ds of th rsoac rgy It s rasoabl to xpct that th lar covrg polyomal wll also b somhow coctd wth th rsoac rgy Zhag t al [78] stablshd a approxmato modl of DRE wth lar covrg polyomal of bzod hydrocarbos: ) RE = z ) (5) =0 For codsd aromatc hydrocarbos wth lar umbr 4 th paramtrs ad ad wghts ( 4) th abov Eq (5) w dtrmd = = 08 = 04 ad = 7 4 = 940 ad = 00 Th corrlato coffct s 0997 ad th ma rror s 004 Gutma t al [8] foud good lar corrlatos btw TRE ad l ( x) for fxd valus of x lyg th trval [0 ] as RE a l ( x) b (6) whr a ad b ar costats Th spcal cas of th abov approxmato (6) for x =0 s th usual approxmato of rsoac rgy va Kulé cout: RE = a0 l K I fact th corrlato coffct R attas a maxmum ad th avrag rlatv rror attas a mmum for som x that cosdrably dffrs from zro Th uxpctd fdg s that th optmal valu of x s always rmarably clos to uty Thr s o statstcally sgfcat dffrc btw th accuracy of th approxmato for optmal x ad for x = Ths lads to th cocluso that () s a quatty of som mportac th lar thory of bzod molculs Furthr studs of Gutma t al [8 86] rvald crta hthrto cocald proprts of rsoac rgs of bzod molculs ad thr dpdc o Kulé- ad lar-structur-basd paramtrs 4 A Extso to Fullrs Smlarly to th sxtt polyomal th lar covrg polyomal ca b aturally xtdd to fullr graphs W also chcd ts rol th stablty of fullrs W Su ad F Wag also computd th lar covrg polyomals of all fullr somrs of ad 70 Thr computato rsults show that (:8) achvs th maxmum () = ad th somr :809 has th scod maxmum () = 5889 Ths ar cosstt wth thr sxtt pattr couts For 70 fullr somrs 70:849 70:776 ad 70:706 hav th frst scod ad thrd maxmum () whch ar ad Tabl Th Frst Scod ad Thrd Maxmum () Fullr Isomrs of ad 70 Isomrs lar covrg polyomal (x) () : x 8x 08x 768x 4956x 4644x 486x : x 06x 08x 6996x 097x 8968x 4656x 09x : x x x 65x 4000x 66900x 67650x 4580x : x 5x 5474x 077x 04690x 067x 7496x 400x 886x : x 5x 4569x 888x 09x 97x 496x 69x 9574x : x 0x 5965x 970x 065x 5x 45490x 585x 94980x

9 Advacs of lar's Aromatc Sxtt Thory Th Op Orgac hmstry Joural 0 Volum rspctvly (s Tabl ) W ca s that th fullr somrs :8 ad 70:849 rta th maxmum valus both sxtt pattr cout ad lar covr cout amog thr fullr somrs Howvr th somrs 70:776 ad 70:706 hav th ar () 's ad ar ust xchagd compard th rag trms of sxtt pattr couts 4 LAR POLYNOMIAL Thr ar varous xtsos of lar structurs Hrdo ad Hosoya [9] allvatd rul (c) th Itroducto to df a (xtdd) lar structur for calculatg rsoac rgs wth mor accuracy: lar structur may cota lss tha cycls but o bzod rg should hav propr or mpropr sxtts El-Basl ad Rad c t al [ 9-95] dscrbd such a xtso of lar structurs ad gav varous costructo approachs For som larg bzod systms howvr thr ar dffrt xplaatos Shu t al [68] clarfd such a xtdd lar structur by rplacg (c) wth (d): th st of crcls s maxmal o w cycl ca b draw usg (a) ad (b) By usg lar covrs a mor prcs graph-thortcal dfto of lar structurs was prstd Lt G b a pla or sphrcal graph wth a prfct matchg A lar covr of G s calld a lar structur f th st of hxagos s maxmal ( th ss of st-cluso) all lar covrs of G ad s calld a propr lar structur f th umbr of hxagos rachs th maxmum all lar covrs of G Th cout polyomal of lar structurs rfrrd to as th lar polyomal was dfd by El-Basl ad Rad c [ ] for a bzod systm G : ( G) ( Gx )= ( Gx ) =0 whr ( G ) dots th umbr of lar structurs of G wth crcls Rad c t al [95] gav a approach to comput th lar polyomal of larg bzods ad obtad th followg rsults: Thorm 4 [95] Lt B b a Kuléa bzod systm Th d x )= hx ) dx h whr th summato gos ovr all hxagos h of B Thorm 4 [95] Lt B b a Kuléa bzod Th x )= hxdx ) h wth th tgratg costat c =0 Shu t al [68] computd th lar polyomal of th cosahdral as follows ( x) = 5x 80x 0 x whch corrcts th rror [67] W ow dscuss a dx cc ) rlatd to lar polyomal of a bzod systm B Lt cc ) dot th umbr of lar covrs wthout altratg hxagos Th ) cc) Th sxtt rotato trasformg all propr sxtts of a Kulé structur of a bzod systm B to mpropr sxtts rsults a drctd tr o th st of Kulé structurs of B wth o root dotd by RB ( ) Lt l ) dot th umbr of o-lavs RB ( ) Thorm 4 [57] Lt B b a bzod systm wth a prfct matchg Th cc ) = l ) Thorm 44 [57] If a bzod systm B has a prfct matchg ad cotas o coro as ts c subgraph th ) = cc) If all lar covrs wthout altratg hxagos a bzod systm ar dtcal to thr lar structurs ) = cc) w ca obta th lar polyomal by umratg lar covrs wthout altratg hxagos I fact th covrs of Thorm 44 dos ot hold For xampl (coro) = cc(coro) So w ca asly obta th lar polyomal of coro as (coro x) = x x x whch of cours agrs wth th arlr rsult of Rf [94] I fact may bzod systms B that cota coro as ts c subgraph wth ) = cc) hav b costructd For mor dtals th radr s rffrd to [57] 5 LINEARLY INDEPENDENT AND MINIMAL ONJUGATED IRUIT POLYNOMIALS OF BENZENOID HYDROARBONS 5 Itroducto Th cougatd crcut modl s a rsoac-thortc modl whch was troducd by Rad c 976 for th study of aromatcty ad cougato polycyclc cougatd systms Eumrato of cougatd crcuts ld to xprssos for th rsoac rgy of polycyclc cougatd hydrocarbos [96] I rct yars varous vstgatos o th cougatd-crcut modl hav b mad [96-04] such as quatum-mchacal ad computatoal aspcts of th cougatd-crcut modl th slcto of th optmum paramtrs of th cougatdcrcut modl ad comparso btw th cougatdcrcut modl ad svral othr modls for computg th rsoac rgs of bzod hydrocarbos I [05] Guo ad Rad c gav a strct dfto of larly dpdt ad mmal cougatd crcuts ad LM -cougatd crcut polyomals as follows Dfto 5 [05] A st S of larly dpdt ad mmal cougatd crcuts of a Kulé structur K of a bzod hydrocarbo B cossts of a maxmum umbr of larly dpdt crcuts of B whch vry crcut

10 96 Th Op Orgac hmstry Joural 0 Volum 5 Zhag t al s a cougatd crcut of K ad has th mmum lgth Dot a crcut of sz 4 S by R ad th summato xprsso of S by RK ( )= R = r ( K ) R r K s th whr ( ) R S = umbr of th crcuts of sz 4 S Th summato xprsso of all sts of LM -cougatd crcuts of all Kulé structurs of B s dotd by RB ( )= R = RK ( )= rr r r K whr = ( ) K = l Th summato xprsso RB ( ) of LM -cougatd crcuts of B s calld LM -xprssos or th LM - cougatd crcut polyomal (smply th LMpolyomal) of B whch s a polyomal of dgr o wth mult-varats ad may also b dotd by a squc of umbrs ( rl r r r ) calld th LM -cod of B Th LM -cougatd crcut polyomals of bzod hydrocarbos play a ctral rol th cougatd-crcut modl bcaus th rsoac rgy RE ) of a bzod hydrocarbo B s smply qual to RB ( )/ KB ( ) Hr K ) s th umbr of Kulé structurs of B Th LM - cougatd crcut polyomals had b also appld to calculat gralzd bod ordrs of polycyclc cougatd hydrocarbos [06] Thus th calculato of th LM - cougatd crcut polyomals of bzod hydrocarbos bcoms a fudamtal problm o th cougatd-crcut modl Howvr for a gral cas th umrato of LM - cougatd crcuts of bzod hydrocarbos rqurs to costruct all Kulé structurs ad th to fd a st of LM - cougatd crcuts for vry Kulé structur Wh th sz of a molcul crass th umbr of Kulé structurs crass fast ad hc umratg LM -cougatd crcuts by ths mthod bcoms tdous Guo ad Rad c [05] vstgatd th proprts ad th costructo of mmal cougatd crcuts of bzod hydrocarbos ad gav th cssary ad suffct codto for a st of cougatd crcuts of a bzod hydrocarbo to b larly dpdt ad mmal Furthrmor thy stablshd som rcursv rlatos for calculatg th LM -cougatd crcut polyomals of svral classs of bzod hydrocarbos so that th LM -polyomals of th svral classs of bzod hydrocarbos ca b drctly obtad from th LM - polyomals ad th Kulé structur couts of thr subgraphs Guo ad Rad c [07] xtdd th rcursv formula for calculatg LM -polyomals for both catacodsd bzod hydrocarbos ad som famls of structurally rlatd prcodsd bzod hydrocarbos Thr ar stll som classs of bzod hydrocarbos whos LM -polyomals caot b obtad by th abov rcursv mthod For gral cass Guo Rad c ad Kl [08] furthr gav a aalytcal xprsso for th cout of LM -cougatd crcuts of B whch s basd o th couts of Kulé structurs of slctd subgraphs of B K By usg th mthod th LM -polyomals of ay bzod hydrocarbo ca b obtad A bzod hydrocarbo H ) s a -coctd pla graph whos vry tror fac s boudd by a rgular hxago A coctd subgraph of a BH s sad to b a BH -fragmt HF ) A -coctd BHF s sad to b a gralzd BH (GBH ) Lt B b a BH or GBH A bod of B s sad to b a fxd bod f t appars always as a doubl bod vry Kulé structur) or always as a sgl bod If B cotas o fxd bod th B s sad to b ormal; othrws B s sad to b sstally dscoctd A ormal compot B of B s a maxmal subgraph of B wth o fxd bod (possbly B = B that s B s ormal) All ormal compots of B ar dotd by B Th boudary of a tror fac of a BH or BH -fragmt B s calld a rg of B Dfto 5 [05] Lt s b a rg ( a hxago) of a bzod hydrocarbo B ad K a Kulé structur of B A cougatd crcut of K (smply a K -cougatd crcut ) s sad to b a mmal cougatd crcut of th rg s f th tror of cotas th tror of s ad has th mmum lgth W also say that a K -cougatd crcut of B s mmal f thr s a rg s B such that s a mmal cougatd crcut of s (s Fg 9) Fg (9) ostructo of mmal cougatd crcuts of a rg s of a bzod hydrocarbo B Thorm 5 [05] Lt K b a Kulé structur of a bzod hydrocarbo B ad lt b a mmal cougatd crcut of a rg s of B Th B [ ] s o of

11 Advacs of lar's Aromatc Sxtt Thory Th Op Orgac hmstry Joural 0 Volum 5 97 th BHs show Fg (9) ad th B [ ] ar uquly dtrmd K doubl bods Thorm 5 [05] Lt K b a Kulé structur of a bzod hydrocarbo B A st S ={ l t} of K -cougatd crcuts of B s a st of LM -cougatd crcuts of K f ad oly f for ay rg s all ormal compots B of B thr s xactly o crcut S such that s a mmal K -cougatd crcut of s 5 Rcursv Mthod for omputg LM -ougatd rcut Polyomals of Bzod Hydrocarbos Thorm 5 stablshs th thortcal bass of th partto of th LM -xprsso of B to th LM - xprssos of rgs of B Th LM -xprsso of a rg s B dotd by Rs ) s dtrmd by tag th summato xprsso of th mmal cougatd crcuts of s o for vry Kulé structur of B ad B RB ( )= Rs ) R ( ) s s B may also b dotd by a squc of umbrs (rg cod) ( rl() s r () s r() s ) whr r () s s th coffct of th trm R Rs ) Thorm 5 also abls us to stablsh som rcursv rlatos for umrato of LM -cougatd crcuts of B Dfto 5 [05] For a dg = uv of a bzod hydrocarbo B lt B ) dot th labld graph of B for whch th dg s labld as doubl (sgl) bod ad Lt B ) dot th ormal compots of Bu v ) [ B ad B may b thought as th ormal compots of B ) sc s fact a fxd doubl (sgl) bod B )] Th subgraph of B ) ducd by th hxagos B ) whch ar ot B ) s dotd by B ') Th cotrbuto of all rgs ) to RB ( ) ( RB ( )) s dotd by R ) B ( R )) ad th cotrbuto of all rgs B ' ' ) to RB ( ) ( RB ( )) s dotd by R ) ( R ) ) larly RB ( ) ad RB ( ) ar ust th LM - xprssos of all th Kulé structurs of B cotag ad ot cotag th dg rspctvly Thus R B R B R B R B R B ( )= ( ) ( ) ( ) ( ) Th abov xprssos gv som parttos of LM - cougatd crcuts of B so that w ca obta RB ( ) from ts all parts Howvr w d to furthr rduc thm to LM -xprssos of subgraphs of B Thorm 5 [05] Lt B B Bt b t mutually dsot BH s or BH -fragmts ad B = BB B t Th t K) RB ( )= RB ( B Bt)= RB ( ) = K ) Thorm 54 [05] Lt B B Bt b th ormal compots of a sstally dscoctd BH or a BH - fragmt B Th K) RB RB RB t ( )= ( )= ( ) = K ) Thorm 55 [05] Lt B b a BH whch cotas o crow (s Fg 0 ()) as ts subgraph Th for ay dg = uv of B ach of LM -cougatd crcuts of B ) s also a mmal B ) Thorm 56 [05] Lt B b a ormal BH whch cotas o crow as ts subgraph Th for ay dg of B RB R B RB RB R B RB ( )= ( ) ( ) ( )= ( ) ( ) RB RB RB R B R B ( )= ( ) ( ) ( ) ( ) Thorm 57 [05] Lt b a dg o th boudary of a bzod hydrocarbo B ad lt S ) ( S ')) b h h Fg (0) () A crow B () A rcursv dg a BH B wth a local structo () Rcursv dgs p a BH B wth a local structo

12 98 Th Op Orgac hmstry Joural 0 Volum 5 Zhag t al th st of rgs B ) Lt ) ( )) dot ' th st of mmal cougatd crcuts of a rg s ) Th for s Sh ) R ( )= ( ) s B K B R ) ( for )/ 4 s s Sh' R ( )= ( ) s B K B R( )/ 4 whr s ) dots th lgth of K ) s th umbr of Kulé structurs of B Dfto 54 [05] Lt b a dg o th boudary of a bzod hydrocarbo B If B ad satsfy o of th followg codtos: () B cotas o crow (s Fg 0 ()) as ts subgraph; () B cotas a local structur as show Fg 0 () ad s th mard dg; () B cotas a local structur as show Fg 0 () ad s th mard dg; th s sad to b a rcursv dg of B Thorm 58 [05] Lt B b a BH whch cotas a rcursv dg o th boudary of B Th R(B)=R (B ) R (B ) 8cm = R(B ) R(B ) cm ss ') ) h s R (B ) R (B ) ss ) h ) s s s K(B )R ( )/4 K(B )R ( )/4 Dfto 55 [05] Lt b a mmal cougatd crcut of a rg s of a bzod hydrocarbo B ad lt s b a hxago of B for whch s ad th tror of s s cotad th xtror of If = s (th symmtry dffrc of dg sts of ad s ) s also a mmal cougatd crcut of s th w say s obtad B from by a xtso ad s s a xtdbl hxago of For a rg s B ) a mmal cougatd crcut of s B ) s sad to b mmum f has th smallst lgth ad B [ ] cotas a smallst umbr of hxagos Thorm 59 [05] Lt b a rcursv dg of a bzod hydrocarbo B ad lt s b a rg B ' ) Lt b a mmal cougatd crcut of s B ) whch s ot mmum Th ca b obtad from aothr mmal cougatd crcut of s B ) by a xtso Procdur 5 [05] Lt b a rcursv dg of a bzod hydrocarbo B ad s a rg B ) Lt b a uqu mmum cougatd crcut of s ) () St S ={ } 0 S = S 0 () For vry mmal cougatd crcut S fd all xtdbl hxagos of xtd to w mmal cougatd crcuts ad st thm to S () If S = th go to (4) Othrws st go to () (4) St s)= S ( s)= S) = = B Fg () A xampl for applcato of procdur

13 Advacs of lar's Aromatc Sxtt Thory Th Op Orgac hmstry Joural 0 Volum 5 99 A xampl of applcato of Procdur 5 s show Fg 5 LM -ougatd rcut Polyomals atacodsd Bzod Hydrocarbos I a gral cas a catacodsd bzod hydrocarbo (cata- BH ) B has th costructo show Fg () whr B B B B5 ar subgraphs of B ach of whch s a cata- BH Partcularly f = = 0 B bcoms a straght cata- BH Thorm 50 [07] Lt B = Ba ( ) dot th straght cata- BH wth hxagos (s Fg (a)) Th ( )= RB ( R ) = Thorm 5 [07] Lt B b a cata- BH show Fg () Th R(B)=R(B ) ( )[K(B )R(B ) K(B )R(B )] K(B )K(B ) ( )R 7 [ K(B =4 )][ = R [ K(B )] = = =0 =0 =6 = =0 R R 5 K B R 5 K B R =45 = =0 =8 = [ ( )] [ ( )] whr f =0 ( =0) th K )= for = 45 ( = 67) K )= 0 for = 890 ( = 45) ad = ( ) = =0 R =0 R = =0 R = =0 orollary 5 [07] Lt B = Bu( m m mt) dot a ubrachd cata- BH as show Fg (4) Th m R(B)=m R(B ) R(B ) [K(B = )(m ) K(B )]R K(B ) m m m R = K(B )[ = R =0 R m R ] K(B 4 ) m m [ R =m R m m m =m R ] K(B 5 )R m m whr f m =0 ( m =0) th K )= for = ( = 4) ad K )= 0 for 4 ( 5) By Thorm 50 ad orollary 5 w ca asly obta th followg calculato formula for umratg th LM -cougatd crcuts of th cata- BH s Fg () orollary 5 [07] Lt Bb ( ) b th cata- BH show Fg (b) Th RB ( ( ))= (6 ) R 4( ) R (4 4 R ) R b = orollary 5 [07] Lt Bc ( ) b th cata- BH show Fg (c) Th R B R R R R R c( c( )) = 4(4 ) 8 4 ( ) 6 = orollary 54 [07] Lt Bd ( ) b th cata- BH show Fg (d) Th R(B d ()) = (8 )R (4 5)R (8 5 8)R R R = orollary 55 [07] Lt B ( ) b th cata- BH show Fg () Th R(B ()) = 96R 8R 8 = (4 5 4)R 6R R R orollary 56 [07] Lt Bf ( ) b th cata- BH show Fg (f) Th RB ( f( ))= F( =0 F R F R F R) whr F = F F s Fboacc s umbr F 0 = F = ad F =0 for ocrg mor gral cass w gv th followg xampls orollary 57 [07] Lt Bu ( m m ) b a ubrachd cata- BH (s Fg 4) Th m R(B u (m m )) = [(m = )(m ) ]R m m m [m = (m ) ]R 4cm R =m m R R R m m = =0 Fg () Th costructo of a catacodsd bzod hydrocarbo orollary 58 [07] Lt Bu ( m m m ) b a ubrachd cata- BH (s Fg 4) Th

14 00 Th Op Orgac hmstry Joural 0 Volum 5 Zhag t al Fg () Som famls of bzod hydrocarbos Fg (4) A ubrachd catacodsd bzod hydrocarbo B RB ( ( m m m )) u m = = {[ m ( m ) ]( m ) ( m )} R m [m = (m )(m ) m m ]R [(m )(m m ) m ]R m = (m ) m m =m m m = =0 R R m m R m = R =0 m m =m m m m R =m (m m )R (m ) R m R m m R m R m m

15 Advacs of lar's Aromatc Sxtt Thory Th Op Orgac hmstry Joural 0 Volum 5 0 R(B h ()) = R(B =0 h ()) [ ( ) = K(B h ()) K(B h ( ))]R 8 K(B = h ())R [ ( ) = K(B h ()) K(B h ( ))]R 4 K B = h R4 K B = h R 5 8 ( ()) 4 ( ) ( ()) whr Fg (5) A catacodsd bzod hydrocarbo BY ( ) cosstg of thr straght catacodsd bzod hydrocarbos orollary 59 [07] Lt BY ( ) dot th cata- BH show Fg (5) Th RB ( ( )) Y = [( = )( )( ) ]R [ ( )( ) ]R = [ = ( )( ) ] R R = =0 =0 R = =0 R = =0 R = =0 R = R R R 54 LM -ougatd rcut Polyomals Som Famls of Structurally Rlatd Prcodsd Bzod Hydrocarbos To umrat LM -cougatd crcuts som famls of structurally rlatd prcodsd bzod hydrocarbos w d to us Thorm 58 ad procdur 5 ad oft d to dal wth svral rcursv rlatos of svral famls of structurally rlatd subgraphs for a famly of structurally rlatd pr- BH s W wll gv som rsults but omt th oprato procsss W frst gv th rcursv formula for umrato of LM -cougatd crcuts of th pr-bhs Fg () orollary 50 [07] Lt Bg ( ) b th BH show Fg (g) Th R(B g ()) = (5 8 5)R = (0 6)R 4( ) R ( )R 6R 4R R orollary 5 [07] Lt Bh ( ) b th BH show Fg (h) Th RB ( (0))= R KB ( ( h h ))= KB ( ( )) KB ( (0))= = h h Kh ( )) = ad Kh ( ))= 0 for ] orollary 5 [07] Lt B ( ) b th BH show Fg () Th for =p ( /) = RB ( ( )) = RB ( ( )) RB ( ( )) (/) [ ( / )K(B = ( )) K(B ( ))]R K(B ( ))R = (/) { ( = )[K(B ( )) K(B ( ))] K(B ( ))}R (/) { [K(B = ( )) ( )K(B ( ))] K(B ( ))}R ; 4 for =p ( )/ =0 RB ( ( )) = RB ( ( )) RB ( ( )) ()/ [ ( = )K(B ( )) K(B ( ))]R K(B ( ))R = ()/ { [( = )K(B ( )) ( ) K(B ( ))] K(B ( ))}R (5)/ { [ = )K(B ( )) K(B ( ))] K(B ( ))}R 4 whr RB ( (0)) = 8 R RB ( ( )) = R K ( /) ( )) = K ( )) K ( )) = for =p ( )/ = K ( )) = K ( )) K ( )) for = p K(0)) = 4 K( )) = K( )) = ad K ( ))= 0 for

16 0 Th Op Orgac hmstry Joural 0 Volum 5 Zhag t al orollary 5 [07] Lt B ( ) b th BH show Fg () Th RB ( ( )) = RB ( ( )) RB ( ( )) RB ( ( )) = [ ( ) K ( )) K ( ))] R = [ K ( )) K ( ))] R 5 = = [ ( ) K ( )) 4 K ( )) 4 K ( 4))] R 5 K B = K B R 4 [ ( ) ( ( )) 4 ( ( 4))] whr RB ( ( ))= 0 for 0 K ( )) = K ( )) K ( )) = K ( ))= for =0 ad K ( ))= 0 for Fally for th two famls of pr- BH s show Fg (6) w gv th rcursv formula for umrato of thr LM -cougatd crcut polyomals orollary 54 [07] Lt B ( ) b th BH show Fg (6()) Th RB ( ( ))=5 RB ( ( )) 4 RB ( ( )) = [4 ( ) K ( )) 5 K ( )) ] R =0 =0 [ (87 8 ) K ( )) K ( )) ] R =0 [4 ( ) K ( )) K ( )) ] R K B =0 K B R4 [ ( ) ( ( )) ( ( )) ] whr =0 R (0))= 0 K ( ))=5 K ( )) 4 K ( )) K (0)) = ad K ( ))= 0 for orollary 55 [07] Lt Bl [ m ] b th BH show Fg (6()) Th RB ( [ m = RB ( [ m RB ( [ m l l l m = K B = l m K Bl R ( [ ( [ whr K[ m = K[ m [ m l l l K[ m0 = K[0 l l 55 Aalytcal Exprssos for th out of LM - ougatd rcuts of Bzod Hydrocarbos Although th LM -polyomals of svral classs of bzod hydrocarbos ca b drctly obtad from th LM -polyomals ad th Kulé structur couts of som subgraphs by th abov rcursv mthod [05 08] thr ar stll som classs of bzod hydrocarbos whos LM xprssos caot b obtad by th rcursv mthod g th bzod hydrocarbos show Fg (7) So w d to vstgat a w mthod to calculat such LM xprssos I rf [08] Guo Rad c ad Kl vstgatd furthr proprts of LM -cougatd crcuts ad dffrt cotrbutos of LM -cougatd crcuts havg dffrt shaps to RB ( ) ad gav a w mthod for calculatg th LM -polyomals of polycyclc bzod hydrocarbos whch s basd o th couts of Kulé structurs of slctd subgraphs of bzod hydrocarbos By usg th mthod th LM -polyomals of ay bzod hydrocarbo ca b obtad Thorm 5 stablshs th thortcal bass of th partto of th LM -polyomal of B to th LM - polyomals of rgs of B Th LM -polyomal of a rg s of a bzod hydrocarbo B dotd by Rs ) s th summato of all th mmal cougatd crcuts of s o for ach Kulé structur of B Not that Rs ) s ust B th cotrbuto of s to RB ( ) ad RB ( )= R) ( Rs )= 0 for s B ) Ad Rs ) may also b dotd by a squc of umbrs (rg cod) ( r() s r() s r () s ) whr r () s s th coffct of th trm R R ) s Hc th cout of LM -cougatd crcuts of a bzod hydrocarbo B s rducd to calculatos of Rs ) for vry rg s of B To calculat Rs ) w d s s Fg (6) Two famls of prcodsd bzod hydrocarbos

17 Advacs of lar's Aromatc Sxtt Thory Th Op Orgac hmstry Joural 0 Volum 5 0 Fg (7) Two famls of prcodsd bzod hydrocarbos to vstgat furthr proprts of mmal cougatd crcuts of bzod hydrocarbos For a Kulé structur K of a bzod hydrocarbo B Thorm 5 gvs all possbl cofguratos of a mmal K -cougatd crcut of a rg s of B (s Fg 9) W call a cofgurato of a mmal cofgurato ( M -cofgurato) wth rspct to (wrt) th rg s ovrsly for a crcut B whch has a M - cofgurato wrt a rg s w hav th followg: Lmma 5 [08] Lt b a cougatd crcut wth a M -cofgurato wrt a rg s a bzod hydrocarbo B Th for ay Kulé structur K B thr s xactly o Kulé structur K B for whch K K ad s a mmal K -cougatd crcut of s B By Lmma 5 w ca gv a w dfto of a mmal cougatd crcut of a rg of B whch s dpdt of a Kulé structur K of B Dfto 56 [08] A cougatd crcut of a bzod hydrocarbo B s sad to b mmal f has a M -cofgurato wrt a rg s B s also sad to b a mmal cougatd crcut of s B orollary 56 [08] Lt b a crcut wth a M - cofgurato wrt a rg s a bzod hydrocarbo B Th corrspods to xactly K ) ( = K B[ ) Kulé structurs for ach of whch s a mmal cougatd crcut of s B Not that orollary 56 caot b usd to calculat Rs ) I gral cass th coffct r () s of th trm R Rs ) s ot qual to K 4 ) hr 4 4 s a mmal cougatd crcut of s wth sz 4 bcaus for a Kulé structur K ad a rg s of B a mmal K -cougatd crcut of s may b ot uqu (s Fg 8) but calculatg Rs ) rqurs o to ta xactly o mmal cougatd crcut of s for ach Kulé structur Thrfor w d to vstgat th proprts of o-uqu mmal cougatd crcuts of a rg B Fg (8) ad ar two mmal rg s K -cougatd crcuts of a Lmma 5 [08] Lt K b a Kulé structur of a bzod hydrocarbo B ad lt ad b two mmal K -cougatd crcuts of a rg s B Th () B [ ] ad B [ ] cota a uqu rg s commo; () ay dg s ( s ) s ot o ( ) ad ay dg se( s) ( se( s) ) s o ( ); () ach compot s a dg (whch s a K -doubl bod); ad (v) thr ar at most two mmal K -cougatd crcuts of s B

18 04 Th Op Orgac hmstry Joural 0 Volum 5 Zhag t al Fg (9) Four typs of pars of mmal cougatd crcuts W call two mmal K -cougatd crcuts of a rg s B as a par of mmal K -cougatd crcuts of s By Lmma 5 w ca classfy pars of mmal cougatd crcuts of a rg s B Thorm 5 [08] Lt ad b a par of mmal K -cougatd crcuts of a rg s a bzod hydrocarbo B for a Kulé structur K of B Th blogs to o of th four typs of pars of mmal cougatd crcuts as show Fg (9) Thorm 5 gvs all possbl cofguratos of a par of mmal K -cougatd crcuts ad of a rg s B W call ths cofguratos PM -cofguratos wrt th rg s ad ad th mutually assocatd mmal cougatd crcuts of s ovrsly for a par of mutually cougatd crcuts ad ( B has Kulé structurs) wth a PM -cofgurato wrt a rg s ad may hav dffrt szs ad ar calld mutually assocatd cougatd crcuts of S Lmma 5 [08] Lt ad b a par of mutually assocatd cougatd crcuts wth th sam sz ad wth a PM -cofgurato wrt a rg s a bzod hydrocarbo B Th for ay Kulé structur K B thr s xactly o Kulé structur K B for whch K K ad both ad ar mmal cougatd crcuts of s B K - By Lmma 5 w ca also gv a w dfto of a par of mmal cougatd crcuts of a rg B whch s dpdt of a Kulé structur K of B Dfto 57 [08] Two mutually cougatd crcuts ad a bzod hydrocarbo B ar sad to b a par of mmal cougatd crcuts of a rg s B f has a PM -cofgurato wrt th rg s ad ad hav th sam sz W also say that ad ar mutually assocatd mmal cougatd crcuts wrt s Rcall th costructo of a mmal K -cougatd crcut of a rg s B show Fg (9) Lt B [ ] b th subgraph of B [ ] ducd by th hxagos labld by 0 abc ad th boudary of B [ ] larly also has a M -cofgurato wrt th rg s ad = = 4( a b c ) = 4 ( R ) W call B [ ] th udrlyg cofgurato of B [ ] ad say that s th udrlyg crcut of ad that has a udrlyg M -cofgurato wrt s For a mmal K - cougatd crcut of a rg s B thr s a uqu crcut wth a udrlyg M -cofgurato wrt

19 Advacs of lar's Aromatc Sxtt Thory Th Op Orgac hmstry Joural 0 Volum 5 05 s for whch B [ ] cotas as may hxagos as possbl so that all tror dgs of B [ ] ar K -sgl bods Th crcut s calld th assocatd crcut of accordg to K Partcularly f all dgs of s ar ot o th boudary of B s calld th odgratd assocatd crcut of (s Fg 0) Th dgs a b ad c dcatd Fg (0) ar calld th xtrm dgs of ; th sts of th tror dgs of B [ ] paralll to a b ad c ar rspctvly dotd by EA EB ad E ; ad E = E { } E = E { } ad E = E { } Not A A a B B b c that f ay o of a b or c s ot o th boudary bb ( ) of B t must b a K -doubl bod I th othr cas t may b a K -sgl or K -doubl bod Th assocatd crcut of s dffrt from a assocatd mmal K - cougatd crcut of sc may b ot K - cougatd ad may b ot qual to Thorm 5 [08] Lt K b a Kulé structur of a bzod hydrocarbo B ; a mmal K -cougatd crcut of a rg s B ; ad th assocatd crcut of accordg to K Th f () a dg o s ls o th boudary bb ( ) of B or () o of th xtrm dgs of ls o bb ( ) ad s a sz tha s a uqu mmal of s B K -sgl bod or () has gratr K -cougatd crcut For a crcut wth a M -cofgurato wrt a rg s B ad th udrlyg crcut of thr s a crcut wth a M -cofgurato wrt s for whch th udrlyg crcut of s ust th tror of s cotad th tror of ad B [ ] cotas as may hxagos as possbl I th cas Fg (()) has th maxmum M -cofgurato calld full M -cofgurato I th cas of Fg (()) s sad to hav a trucatd M -cofgurato W call th charactrstc crcut of ad For a mmal K -cougatd crcut of a rg s ad th assocatd crcut of accordg to K a charactrstc crcut of such that has a PM -cofgurato wrt s s also calld th assocatd charactrstc crcut of If a charactrstc crcut of has a trucatd M -cofgurato for th dgs f f ft o whch l o th boudary bb ( ) of B ad whos d vrtcs hav dgr thr B [ ] lt F dot th st of th dgs B [ ] whch cota f ad ar trsctd by a sam l sgmt L (s Fg ()) W call F = t th charactrstc dg sts of B [ ] Thorm 54 [08] Lt b a mmal K - cougatd crcut of a rg s of a bzod hydrocarbo B for a Kulé structur K of B ; th odgratd assocatd crcut of accordg to K wth l ; ad th assocatd charactrstc crcut of Th s a uqu mmal K -cougatd crcut of s f ad oly f thr a xtrm dg of ls o th boudary of B ad s a K -sgl bod or ad thr s a charactrstc dg st ach dg F s a has a trucatd M -cofgurato K -doubl bod F of B [ ] such that By Thorms 5 ad 54 w d oly to cosdr th bzod hydrocarbos wth o fxd bod To calculat Rs ) t s ough to gv a mthod to dtrm th coffct r () s of th trm R Rs ) for From a Kulé structur K ad a mmal K - cougatd crcut of a rg s B w hav troducd th cocpts of th assocatd crcut ad th assocatd charactrstc crcut of accordg to K Now w d Fg (0) Th assocatd crcut of a mmal K -cougatd crcut of a rg s B accordg to K

20 06 Th Op Orgac hmstry Joural 0 Volum 5 Zhag t al Fg () () A charactrstc crcut of wth a full M -cofgurato () A charactrstc crcut of wth a trucatd M - cofgurato whr f f ft l o th boudary of B ad thr d vrtcs hav dgr thr B [ ] to gralz th cocpts so that thy ar dpdt of a Kulé structur of B Dfto 58 [08] For a par of mmal cougatd crcuts ad of a rg s a bzod hydrocarbo B ad thr udrlyg crcuts ad s sad to hav a udrlyg PM -cofgurato wrt s For a mmal cougatd crcut of s ad ts udrlyg crcut a crcut s sad to a odgratd assocatd crcut of ad f has a PM - cofgurato wrt s has a udrlyg PM - cofgurato wrt s ad ay dg of s s ot o th boudary of B A charactrstc crcut of s sad to b a assocatd charactrstc crcut of ad f has a PM -cofgurato wrt s For a mmal cougatd crcut of a rg s B lt A ( )={ s a odgratd assocatd crcut of } A( )={ A( ) ad < } A( )={ A( ) ad = } A ( )={ s a assocatd charactrstc crcut of } A( )={ A ( ) < ad trucatd M -cofgurato } A( )={ A ( ) = ad trucatd M -cofgurato } has a has a For calculato of Rs ) th pars of crcuts wth udrlyg PM -cofguratos play a y rol Fg () shows four typs of pars of crcuts wth udrlyg PM - cofguratos wrt a rg s For a par of crcuts ad B wth a udrlyg PM -cofgurato w dot by (rspctvly charactrstc crcut of (rspctvly ) th ) whch blogs to A ( ) [rspctvly A ( ) ] whr th crcut (rspctvly ) s sad to b of typ ( {4}) f th par of crcuts s of typ A mmal cougatd crcut of a rg s s sad to b of typ f th udrlyg crcut of s somorphc to o of crcuts ad of typ For covc w troduc th followg otatos: () ()={ s s a crcut a par of crcuts ad wth a udrlyg P -cofgurato wrt a rg s B whch ar of typ ad = 4 } r () () s : th umbr of all th crcuts dotd by R Rs ) whch ar of typ = 4 K[ : th umbr of th Kulé structurs of B for whch all dgs ad all xtrm dgs ( a p ) of ad ar doubl bods ad all tror dgs of B [ ] ad B [ ] ar sgl bods K[ : th umbr of th Kulé structurs of B for whch all xtrm dgs of ad ad all dgs ad a charactrstc dg st of B [ ] ar doubl bods ad all tror dgs of B [ ] ad B [ ] ar sgl bods K[ E E : th umbr of th Kulé structurs of B for whch all xtrm dgs of ad th dgs