Counting Polynomials and Related Indices by Edge Cutting Procedures

Transcription

1 MATCH Commuicatios i Mathmatical ad i Computr Chmistry MATCH Commu. Math. Comput. Chm. 64 (010) ISSN Coutig Polyomials ad Rlatd Idics by Edg Cuttig Procdurs M. V. Diuda Faculty of Chmistry ad Chmical Egirig, Babs-Bolyai Uivrsity, 40008, Cluj, Romaia (Rcivd Jauary 13, 010) Abstract. A topological idx is a umric quatity drivd from th structur of a graph G(V,E) which is ivariat up to automorphisms of th cosidrd graph. O of th most famous topological idics is th Wir idx W(G); it quals th sum of distacs btw all uordrd pairs of vrtics of G. A rlatd umbr is th Szgd idx SZ(G), which is th sum of all products of o-quidistat, proximal vrtics u (), v () with rspct to th two ds of ay dg =(u,v) i G. Third is th Cluj idx CJ S(G), calculatd from th first drivativ of CJ (x) polyomial. A forth idx, calld Cluj-Ilmau CI(G), is calculatd from th first ad scod drivativs of th Omga ( x) polyomial, which couts th opposit dg strips i G. All ths idics ad rlatd polyomials ar drivd hr by dg cuttig procdurs i som bipartit graphs ad/or partial cubs. A clar rlatdss amog ths dscriptors was stablishd ad xmplifid. Thir us i corrlatig various physico-chmical or biological proprtis with th molcular structur hav b xtsivly prov. 1. Itroductio O of th most famous topological idics is th Wir idx, itroducd by Harold Wir. 1 Th Wir idx quals th sum of topological distacs btw all uordrd pairs of vrtics of G: WG ( ) d ( uv, ) (1) ( uv, ) V( G) G Th Szgd idx is aothr topological idx dfid by Iva Gutma as: Sz ( G ) u() v() () ( uv, ) E( G) whr u () is th umbr of vrtics of G lyig closr to u tha to v ad v () is th umbr of vrtics of G lyig closr to v tha to u.

2 -570- W proposd Cluj matrics ad idics i viw of xtdig th dfiitio of Wir matrics, proposd by Radi 3,4 to cycl-cotaiig graphs, othr tha th Szgd idx did. A Cluj fragmt 5-9 CJ i, j, p collcts vrtics v lyig closr to i tha to j, th dpoits of a path p(i,j). Such a fragmt collcts th vrtx proximitis of i agaist ay vrtx j, joid by th path p, with th distacs masurd i th subgraph D (G-p) : CJ v v V ( G); D ( i, v) D ( j, v) (3) i, j, p ( Gp) ( Gp) I trs, CJ i, j, p dots sts of (coctd) vrtics v joid with j by paths p goig through i. Th path p(i,j) is charactrizd by a sigl dpoit, which is sufficit to calculat th usymmtric matrix UCJ. I graphs cotaiig rigs, th choic of th appropriat path is quit difficult, thus that path which provids th fragmt of maximum cardiality is cosidrd: [UCJ] i, j max CJ p i, j,p (4) Wh path p blogs to th st of distacs DI(G), th suffix DI is addd to th am of matrix, as i UCJDI. Wh path p blogs to th st of dtours DE(G), th suffix is DE. Wh th matrix symbol is ot followd by a suffix, it is implicitly DI. Th Cluj matrics ar dfid i ay graph ad, xcpt for som symmtric graphs, ar usymmtric ad ca b symmtrizd by th Hadamard (pair-wis) multiplicatio 10 with thir trasposs: SM p = UM (UM) T (5) If th matrics calculatd o dgs (i.., o adjact vrtx pairs) ar rquird, th matrics calculatd o paths must b multiplid by th adjaccy matrix A (which has th o-diagoal tris of 1 if th vrtics ar joid by a dg ad, othrwis, zro): SM = SM p A (6) Th Cluj idics, calculatd as half sum of th matrix tris, prviously usd i corrlatig studis publishd by TOPO GROUP Cluj, wr calculatd i th symmtric matrics, thus ivolvig a multiplicativ opratio. Also, th symbol CJ

3 -571- (Cluj) is usd hr for th prviously dotd CF (Cluj fragmtal) matrics ad idics. I this papr, th usymmtric matrix dfid o distacs ad calculatd o dgs UCJ will b usd to compar th cofficits of th Cluj polyomials 11,1 with thos obtaid by a cuttig procdur (s blow): UCJ = UCJ p A (7). Basic dfiitios Lt G(V,E) b a coctd bipartit pla graph, with th vrtx st V(G) ad dg st E(G). Two dgs =(x,y) ad f=(u,v) of G ar i rlatio opposit, op f, if thy ar opposit dgs of a fac i G. Assumig that facs ar isomtric subgraphs of G, th rlatio op implis th coditio of topologically paralll dgs : 13 d( x, v) d( x, u) 1 d( y, v) 1 d( y, u) (8) Rlatio op is rflxiv ad symmtric but, i gral, is ot trasitiv. It will partitio th dgs st E(G) ito opposit dg strips ops, S(G)={ S 1, S,..., S }, as follows. (i) Ay two subsqut dgs of a ops ar i op rlatio; (ii) Ay thr subsqut dgs of such a strip blog to adjact facs; (iii) Th ops is ta as maximum possibl, irrspctiv of th startig dg. (iv) Th choic about th maximum siz of fac/rig, ad th fac/rig mod coutig, will dcid th lgth of th strip. Thr ar graphs i which op is trasitiv ad ops suprimpos ovr th orthogoal cut strips ocs, C(G)={ c 1, c,..., c }, dfid by rlatio co. 13,14 I such a graph, rlatio op, dfid locally (o facs), bcoms a global proprty, li th corlatio ad th graph is a co-graph or a partial cub. Th its orthogoal cuts form a partitio of th dgs i G: E( G) c1c... c, ci cj, i j. Th ocs ca b obtaid by a orthogoal dg-cuttig procdur (s blow). A subgraph H G is calld isomtric, if d ( u, v) d ( u, v), for ay ( uv, ) H; it is covx if ay shortst path i G btw vrtics of H blogs to H. A graph G is a partial cub if it is mbddabl i th -cub Q, which is th rgular graph whos vrtics ar all biary strigs of lgth, two strigs big H G

4 -57- adjact if thy diffr i xactly o positio. 15 Th distac fuctio i th -cub is th Hammig distac. A hyprcub ca also b xprssd as th Cartsia product: Q K i 1 For ay dg =(u,v) of a coctd graph G lt uv dot th st of vrtics lyig closr to u tha to v: w V( G) d( w, u) d( w, v) uv. It follows that wv( G) d( w, v) d( w, u) 1. Th sts (ad subgraphs) iducd by ths uv vrtics, uv ad vu, ar calld smicubs of G; th smicubs ar opposit ad disjoit os. 16,17 A graph G is bipartit if ad oly if, for ay dg of G, th opposit smicubs dfi a partitio of G: v V( G). Ths smicubs ar just th vrtx uv vu proximitis of (th dpoits of) dg =(u,v), which th Cluj polyomials cout (s blow). Th rlatio co is rlatd to ~ (Djoovi 18 ) ad (Wilr 19 ) rlatios: 0 i a coctd bipartit graph, co = ~ =. For two dgs =(u,v) ad f=(x,y) of G th thta rlatio is dfid as: f if d( u, x) d( v, y) d( u, y) d( v, x). A bipartit graph G is a co-graph if ad oly if it is a partial cub, ad all its smicubs ar covx; rlatio co / is th trasitiv. 17 A co-graph ca also b o-bipartit. 3. Cluj ad rlatd polyomials by th cuttig procdur Th Cluj polyomials ar dfid 11,1,1, o th basis of Cluj matrics as: CJ ( x) m( ) x (9) Thy cout th smicub or proximity p of th vrtx i with rspct to ay vrtx j i G, joid to i by a dg {p,i } (th Cluj-dg polyomials) or by a path {p p,i } (th Cluj-path polyomials), ta as th shortst (i.., distac DI) or th logst (i.., dtour DE) paths. I q. (9), th cofficits m() ca b calculatd from th tris of usymmtric Cluj matrics by th TOPOCLUJ softwar program. 3 Th summatio rus ovr all {p} i G. I bipartit graphs, th smicubs coutd by CJ polyomial ca b stimatd by a orthogoal dg-cuttig procdur.,4-6 To prform it, ta a straight li

5 To ay orthogoal cut c, =1,,.., max two umbrs ar associatd: first o rprsts th umbr of dgs itrsctd or th cuttig cardiality c whil th scod (i roud bracts, i Figur 1) is v or th umbr of poits lyig to th lft had with rspct to c. Out of CJ polyomial, thr ar othr topological dscriptors that cout th smicubs i G (s Figur 1, th polyomial xpots), thy diffrig oly i th mathmatical opratio usd i r-composig th dg cotributios to th global graph proprty. Bcaus th opposit smicubs dfi a partitio of vrtics i a bipartit graph, it is asily to idtify th two smicubs: uv = v ad vu = v-v or vicvrsa sgmt, orthogoal to th dg, ad itrsct ad all its paralll dgs (i a polygoal pla graph). Th st of ths itrsctios is calld a orthogoal cut (oc for short) of G, with rspct to (Figur 1). CJ S(x) = 3 3(x 5 +x 11 )+ 3 6(x 16 +x 110 )+ 3 8(x 31 +x 95 )+ 3 8(x 47 +x 79 ) (x 63 +x 63 ) CJ S (1) = 194; CJ S (1) = PI v (x) = 3 3(x )+ 3 6(x )+ 3 8(x )+ 3 8(x ) (x ) PI v (1) = 194; PI v (1) = CJ P(x) = 3 3(x 5 11 )+ 3 6(x )+ 3 8(x )+ 3 8(x ) (x ) CJ P (1) = W(x) = 3 (x 5 11 )+ 3 (x )+ 3 (x )+ 3 (x )+ 3 1(x ) W (1) = H`(1) = ( x) = 3 x 3 +3 x 6 +( )x 8 = 6x 3 +6x 6 +15x 8 CI( G) 9046 Figur 1. Cuttig procdur i th calculus of svral topological dscriptors

6 (ii) Pair-wis summatio, with th polyomial calld (vrtx) Padmaar- Iva 8 by Ashrafi 9-3 (ad symbolizd PI v ): Th cofficits of ths dscriptors ar calculatd (with som xcptios) as th product of thr umbrs (i th frot of bracts - right had part of Figur 1) with th maig: (i) symmtry of G; (ii) occurrc of c (i th whol structur) ad (iii). Rsumig to th mathmatical opratio usd i r-composig th graph smicubs, four polyomials ca b dfid accordig to: (i) Summatio, ad th polyomial is calld Cluj-Sum, by Diuda t al. 11,1,1,,7 (ad symbolizd CJ S): v v v CJ S( x) x x (10) PI ( x) x v v( vv ) (11) (iii) Pair-wis product, whil th polyomial is calld Cluj-Product (ad symbolizd CJ P) 5-9,,6 or also Szgd polyomial (ad symbolyzd SZ): 30-3 CJ P( x) SZ( x) x v ( vv ) (1) (iv) Sigl dg pair-wis product ad th polyomial is calld Wir W(x): W( x) x v ( vv ) (13) Th first drivativ (i x=1) of a (graph) coutig polyomial provids sigl umbrs, oft calld topological idics. Som commts ar ow wlcom. It is ot difficult to s that th first drivativ (i x=1) of th first two polyomials givs o ad th sam valu, howvr, thir scod drivativ is diffrt (s Figur 1) ad th followig rlatios hold i ay graph: 1 CJ S(1) PI (1) ; CJ S(1) PI (1) (14) v Th umbr of trms, giv by CJ S(1)= is twic th umbr giv by PI v (1) bcaus, i th last cas, th two dpoit cotributios ar pair-wis summd for ay dg i a bipartit graph (s (10) ad (11)). It is ot difficult to obsrv th first drivativ (i x=1) of PI v (x) tas th maximal valu i bipartit graphs: v

7 whr u,v, v,u cout th o-quidistat vrtics with rspct to th dpoits of th dg =(u,v) whil m(u,v) is th umbr of quidistat vrtics vs. u ad v. Howvr, it is ow that, i bipartit graphs, thr ar o quidistat vrtics vs. ay dg, so that th last trm i (16) will miss. Th valu of PI v (G) is thus maximal i bipartit graphs, amog all graphs o th sam umbr of vrtics; th rsult of (16) ca b usd as a critrio for chcig th bipatity of a graph. Th third polyomial uss th pair-wis product; otic that Cluj-Product CJ P(x) is prcisly th (vrtx) Szgd polyomial SZ v (x), dfid by Ashrafi t al This coms out from th rlatios btw th basic Cluj (Diuda 5,34,35 ) ad Szgd (Gutma,35 -s rlatio ()) idics: PI (1) v E( G) V ( G) (15) v It ca also b s by cosidrig th dfiitio of th corrspodig idx, as writt by Ili: 33 PI ( G) PI (1) V E m v v uv, vu, uv, uv uv (16) CJ P(1) CJ DI( G) SZ( G) SZ (1) (17) v All th first thr polyomials (ad thir drivd idics) do ot cout th quidistat vrtics, a ida itroducd i Chmical Graph Thory by Gutma. Th last polyomial w call Wir, bcaus it is calculatd as Wir did i calculatig th idx W(G) i tr graphs: multiply th umbr of poits lyig to th lft ad to th right of ach dg (actually rad orthogoal cut c ): WG ( ) W(1) v ( vv) (18) whr v ad v-v ar th cardialitis of th disjoit smicubs formig a partitio with rspct to ach dg i c ta, howvr, as a sigl dg (as i trs). I fact, th rlatio (18) couts paths xtral to th orthogoal cuts c, as th Wir matrix W, proposd by Radi, dos. Th both dscriptors ar rstrictivly dfid: oly i trs (th matrix W) ad oly i partial cubs (th polyomial W(x)). Not that tr graphs ar partial cubs. Th both abov dscriptors cout vrtics (ot dgs). I th opposit, th Hosoya polyomial H(x) couts dgs (ot vrtics), by worig o th Distac D matrix: 15,35 H( x) m( ) x (19)

8 -576- whr th xpot dots th shortst paths (btw pairs of vrtics i G) of xtt, whil m() couts th umbr of -paths. Th dfiitio of W(G), as giv i rlatio (1), is thus rlatd to th (first drivativ H`(1) of ) Hosoya polyomial. Clarly, th both polyomials will provid th sam valu of W(G) i trs/partial cubs, accordig to th thorm of Kli, Gutma ad Luovits, 41 which stats th quality of sums of th itral paths (collctd by D &D p matrics) ad th xtral paths (giv by W &W p matrics): 4 WG ( ) W(1) H(1) (0) Klavžar 5 statd that, i calculatig th idx W(G), th orthogoal cut procdur is applicabl oly i partial cubs. Thus, w ca writ th followig Propositio 1: A bipartit graph i which th rlatio (0) holds is a partial cub. From th abov discussio, th propositio appars at last cocivabl. Mor ovr, th uppr bod of th products i rlatio (18) is rachd for v =v/ ad th umbr of ths maximal lgth ocs is limitd by th symmtry of G. Thus, a graph i which th followig iquality holds is ot a partial cub: 6 WG ( ) SG ( ) ( v/) (1) Howvr, a valu of W(G) lowr tha th abov bod ad, dos ot sur G is a partial cub. I such a cas, tryig to prform th cuttig procdur, a valu v >v/ will idicat a o-covx, o-isomtric subgraph ad thus a graph which is ot a partial cub. Th fial proof is th chcig of trasitivity of co-rlatio. A last rmar o W(x): i partial cubs, its xpots ar idtical to thos i CJ P(x) =SZ(x) whil th cofficits ar thos i th abov polyomials, dividd by. 4. Omga ad rlatd polyomials by th cuttig procdur Lt s ow rtur to Figur 1 ad itroduc th last dscriptor: th Omga polyomial. Dot by m(s) or simply m th umbr of ops of lgth s= s ad dfi th Omga polyomial as: 16,17,43-5

9 -577- s ( x) ms ( ) x () s Th xpots cout just th itrsctd dgs by th cut-li, which is ot dd to b orthogoal o all th dgs of a ops (s abov); th cofficits m(s) ar asily coutd from th symmtry of G. I partial cubs, othr two rlatd polyomials 16,17 ca b calculatd o ops: s ( x) ms x (3) s s ( x) ms x (4) s Th (x) couts quidistat dgs whil (x) o-quidistat dgs. Thus, Omga ad its rlatd polyomials cout dgs ot vrtics. Thir first drivativ (i x=1) provids sigl umbr topological dscriptors: (1) m s E ( ) s (5) (1) m s ( ) s (6) (1) ms ( s ) ( G ) (7) s O Omga polyomial, th Cluj-Ilmau idx, 13 CI=CI(G), was dfid: { } CI( G) [ (1)] [ (1) (1)] (8) A polyomial rlatd to (x) was dfid by Ashrafi 53 as: PI ( x) x u (, ) v (, ) (9) E( G) whr (,u) is th umbr of dgs lyig closr to th vrtx u tha to th v vrtx. Its first drivativ (i x=1) provids th PI(G) idx proposd by Khadiar. 8,54 Propositio. I co-graphs/partial cubs, th quality CI( G) ( G) holds. This ca b dmostratd by xpadig dfiitio (8), CI calculatio ladig to ( G) : 16,17 CI( G) ms m s m s ( s 1) m s ( G) s s s (30) s Rlatio (30) is valid oly i th assumptio c s, which provids th sam valu for th xpot s ad this is prcisly achivd i co-graphs/partial cubs.

10 -578- A graph, of which ( x) ca b writt xactly i th trms of ( x), accordig to th pair rlatios {()&(3)}, will prcisly show th quality CI( G) ( G) cf (30). Th rlatdss of th two polyomials (ad idtity CI( G) ( G) ) is providd rathr by th quality of cardialitis s c tha by th corrspodig sts suprpositio s c, th coditio {()&(3)} big thus cssary but ot sufficit i ordr a graph to b dclard co-graph/partial cub. Fially, th trasitivity of ops/ocs must b prov. Not that thr is ot ow a simpl procdur to stablish th partial cub status. 17 Th quality CI( G) ( G) ca appar v th pair rlatios {()&(3)} ar ot rlatd. This is bcaus th quidistac rlatio qd ivolvs both coditios for topologically paralll (rlatio (8)) ad prpdicular (rlatio (31)) dgs: d( u, x) d( u, y) d( v, x) d( v, y) (31) I such a cas, th idx quality ca b cosidrd as a cas of dgracy. If th graph is co-graph/partial cub, th all of its smicubs ar covx. 17,0 Furthr, a orthogoal dg-cuttig procdur ca b usd to gt th ops. I gral, ( G) PI( G), th diffrc btw th two idics origiatig i th diffrt dfiitio (Ashrafi 44 ) of dg distac: th distac from a vrtx z to a dg ( u, v) is ta as th miimum distac btw th giv poit ad th two dpoits of : d(,) z mi{(, d z u), d(,)} z v (3) Th, th dg =(u,v) ad f=(x,y) ar i rlatio qd f if: d( x, ) d( y, )ad d( u, f) d( v, f) (33) Rlatios (8)&(31) ar strogr tha rlatios (3)&(33), i bipartit graphs thy suprimposig to ach othr (but ot i gral graphs) ad ( G) PI( G). Sic ay partial cub is also a bipartit graph, th i partial cubs/co-graphs th followig tripl quality holds: 16,17,6 CI( G) ( G) PI ( G) (34) I th opposit, i gral graphs, th quality chags to th corrspodig iquality: CI( G) ( G) PI ( G) (35)

11 -579- Rsumig, th status of co-graph/partial cub caot b dcidd by a simpl ad rapid critrio/coditio. Out of various algorithms proposd to rach this tas, th tstig of trasitivity of ocs is th last proof. To rduc th umbr of graphs tstd, th coditios {(0)&(30)}, ca b cosidrd, udr th rsrv thy ar cssary but ot sufficit. 5. Applicatios I th followig, w apply th cuttig procdur o two classs of structurs: (i) pcu cubic t ad (ii) topological aocos. Formulas ar giv symbolically, i viw of asily udrstadig th cuttig procdurs (i associatio with th graphs i figurs) ad oly i fial, at th first drivativ calculatio, th t paramtr ar substitutd. Numrical xampls ar giv Cuttig procdur i pcu cubic t W apply ow th orthogoal cuttig procdur i th pcu cubic twor, apparig i crystal structur (Figur ); amog various ocs, th ctral o is dotd by =0. Th formulas for th t paramtrs ad topological dscriptors ar giv i Tabl 1. C(3,3,3) C =1 C =0 Figur. Cuttig procdur i pcu cubic t Tabl 1. Nt paramtrs ad topological dscriptors i pcu cubic lattic. Typ Formulas vca ( ( )) 3 vca ( ( )) V( Ca ( )) ( a 1) Ca ( ( )) Ca ( ( )) EG ( ) 3 aa ( 1) v v ( C( a)) ( a 1)

12 -580- v v C a a s ( ( )) ( 1) / ( C( a)) ( a 1) Wir ( a1)/ (( v/) v( vv ) W ( C( a, odd), x) 3x 6 x 1 3 (( a1) /) WCaodd ( (, ), x) 3x ( a1)/ 3 ( a1) [( a1) ( a1) ] 6 x 1 ( a1)/ 3 3 W( C( a, odd),1) 3[( a1) /] 6 ( a1) [( a1) ( a1) ] a/ W ( C( a, v), x) 6 x 1 a/ W ( C( a, v), x) 6 x 1 v ( vv ) 1 3 ( a1) [( a1) ( a1) ] ( (, ),1) a/ 6 ( 1) [( 3 1) ( 1) ] 1 W C a v a a a 5 W( C( a),1) (1/) a( a)( a 1) Exampls: a=4; W(x) = 6x x 3750 ; W '(1) = a=5; W(x) = 6x x x ; W '(1) = Szgd SZCa ( ( ), x) ( Ca ( )) WCa ( ( ), x) ( a1) WCa ( ( ), x) 7 SZ(1) ( a 1) W ( C( a)) (1 / ) a( a )( a 1) Exampls: a=4; SZ(x)=150x x 3750 ; SZ' (1) = a=5; SZ(x)=16x x x ; SZ '(1) = Cluj ( a1)/ v/ v vv CJ S CJ S( C( a, odd), x) 6 [ x ( x x )] 1 3 ( a1) / CJ S( C( a, odd), x) 6( a 1) [ x ( a1)/ ( a1) ( x 3 [( a1) ( a1) ] x )] 1 a/ v v v CJ ( (, ), ) 6 ( S C a v x x x ) 1 a/ a ( 1) [( a1) 3 a ( 1) ] CJS( C( a, v), x) 6( a 1) ( x x ) 1 a v CJ ( ( ), ) 6 S C a x x 1 a a ( 1) CJS( C( a), x) 6( a 1) x CJS(1) v 3 a( a 1) ( a 1) 3 a( a 1) Exampls: a=4; CJ S(x)=150x x x x 5 ; CJ S '(1)= a=5; CJ S(x)=16x x x x x 36 ; CJ S '(1)=

13 -581- Omga ( b1)( c1) ( a1)( c1) ( a1)( b1) (C( abc,, ), x) ax bx c x ( a1)( c1) ( a1) (C( aac,, ), x) ax c x ( a 1) (C( aaa,, ), x) 3a x ( Ca ( ),1) 3 aa ( 1) ( Ca ( ),1) 3 a( a1) ( a ) 4 CI( C( a)) 3 a(3a 1)( a 1) Exampls: 5 a=4; ( x) 1 x ; (1) 300; CI a=5; ( x) 15 x ; (1) 540; CI Cuttig procdur i aocos Coical ao-structurs hav b rportd i Naoscic sic 1968, 55,56 bfor th discovry of fullrs. If a graphit sht is dividd ito six sctors, ach with a agl of 60º, ad if m of ths sctors (with m=1 to 3) ar dltd squtially, th daglig bods big fusd togthr, thr classs of graphs, associatd to sigl-walld aocos, ar obtaid; thir apx polygo will b a ptago (a=5), a squar (a=4) or a triagl (a=3), rspctivly. O ca xtd th costructio pricipl ad accpt i th family of topological cos structurs havig th apx polygo a 6 ; of cours, that co with a=6 is just th pla graphit sht whil thos havig largr polygos will show a saddl shap. I th rct yars, svral rsarchrs hav cosidrd th mathmatical proprtis of such aostructurs Figur 3 givs thr xampls of such topological cos, with th applicatio of th cuttig procdurs i viw of drivig som importat topological dscriptors. Figur 3. Cuttig procdur i aocos of apx a=4,6 ad 8

14 -58- Formulas, rfrrig to t paramtrs ad dscriptors ar giv i th Tabls ad 3, alog with som umrical xampls, i Tabl 4. Tabl. Nt paramtrs ad topological dscriptors i bipartit (partial cubs) aocos Typ Formulas for Cos C(a,); a=v; a>4. va (, ) va (, ) a ( 1) a (, ) a (, ) ( a/ )(3 5 ) h h h 0 h0 1 v v (i1) () i1 s ( 1) CJ S(x) CJS( x) CJS0( x) CJS( x) v/ v/ CJS0( x) [( a / )( h0 1) ( a 6)( 1)] ( x x ) v/ v/ CJ S ( x) ( a /) ( x x ) 0 0 CJ S ( x) a ( h v 1) ( x v v x ) 1 CJ S ( x) a v ( x v v x ) 1 v/ v/ ( ) ( /) 0 ( ) ( v v v) 1 3 ( 1) ( ) a v ( / )(3 5 ) 3 v (1) (1) ( / )( 1) (3 ) ( v/) ( ) ( /) 0 ( ) v( vv) (1) ( / 4)( 1) ( CJ S x a x x a x x CJ S(1) v ( a / )(3 5 ) a( 1) ( a / )( 1) (3 ) PI v (x) PI x x a x PI CJ S v a CJ P(x) CJ P x a x a x CJ P a a a a 1a 19 3a 3a 6 aa ) CJP(, x) s W ( x) Wir ( v/) v( vv) W( x) ( a/)( x ) a x 1 (1) ( /)( /) ( ) ( a,,1) (1/10) ( 1)( W a v a v v v W C a a a 15a a 45a 45a76 140a 15 a )

15 ( s,,1) (1/15) ( 1)( W C s s s s 45s 0s 1945s70s15 s) W C 4 3 ( 3,,1) (1/ 5)( 1)( ) Omga 1 ( 1) ( x) ( a/) x ax (1) ( a/ )(3 5 ) CI a a a a a a 3 ((, ) (1/1) ( 1)( ) Tabl 3. Nt paramtrs ad topological dscriptors i bipartit (o-partial cubs) aocos Typ Formulas for Cos C(4,). v(4, ) v(4, ) 4( 1) (4, ) (4, ) (3 5 ) s 1 v v ( ) =odd: =v: last ormal cut 0 ( 1)/ 0 / corrctd cut c 0 1 ( 3)/ c 0 1 ( )/ corrctio c ( c) 1 c ( c) 1 1 CJ c S(x) v/ v/ v CJ S( C(4, ), x) 4( 1)( x x ) 4 ( x vv x ) 1 v vv v( c) 4 ( c) ( x x ) 4 c ( x vv( c) x ) c c CJ S( C(4, ),1) v (3 5 ) 4( 1) 8(3 )( 1) 3 CJ P(x) c 1 ( v/) CJ P( C(4, ), x) 4( 1) x 4 v( vv) x 1 v( vv) ( v( c) )( vv( c) ) 4 ( c) x 4c x c c 3 ( (4, v),1) 16 (538 / 5) (419 /15) 370 CJ P C (1669 / 6) (557 / 5) (557 / 30) ( (4, odd ),1) (31/ ) (53 / 5) (813 / 30) 370 CJ P C Omga 1 ( 1) ( x) ( a/) x ax (1669 / 6) (557 / 5) (557 / 30) (1) ( a/ )(3 5 ) CI a a a a a a 3 ((, ) (1/1) ( 1)( ) As ca b s from Tabls ad 3, Omga polyomial is calculatd by th sam gral formula i ay cos with a 4, a=v.

16 -584- It is importat to s that if G allows a orthogoal cut th { } { c } { s }; howvr, i cos with a=4 ad a=odd, a dtail o ths sts is dd. I all cos, with a 4, a=itgr, th quality CI( G) ( G) holds, by th followig rasos: (i) cos with v a 4 ar partial cubs; (ii) cos with odd a 4 ar uios of partial cubs i o-bipartit graphs ({ c } { s } ad co is trasitiv, thus th cos ar co-graphs, but ot partial cubs); (iii) cos with a 4 show s c (but { c } { s }, c is o-trasitiv ad th bipartit graphs ar ot co-graphs or partial cubs), th last cas big cosidrd as a cas of dgracy. I cos with a=3, CI( G) ( G) bcaus s c (ad th o-bipartit graphs ar ot co-graphs or partial cubs). I ay co with a=v (i.., bipartit graphs), th quality ( G) PI( G) holds. Tabl 4. Exampls for th formulas i Tabls ad 3. a Polyomial Idx 4 3 CJ S(x) = 0x x x x x 3 + 1x x 9 + 4x 0 + 0x 9 PI v (x)= 88x 64 SZ v (x)=0x x x x x 104 ( x) =4x 5 +4x 6 +4x 7 +x CJ S(x) = 4x x x 6 + 4x x x x x x x x 38 +8x 4 + 4x 11 PI v (x)= 140x 100 SZv(x)=4x x x x x x x 500 ( x) =4x 6 +4x 7 +4x 8 +4x 9 +x CJ S(x)=30x x x x x x x 9 PI v (x)= 13x 96 SZ v (x)=30x x x W(x)=6x x x ( x) =6x 5 +6x 6 +6x 7 +3x CJ S(x)=36x + 4x x + 54x x x x x x 11 PI v (x)= 10x 150 CJ S (1)= 563 PI v ' (1) = 563 SZ v '(1)=7590 CI=7176 CJ S'(1)= PI v '(1) = SZv'(1)= CI= CJ S'(1)=167 PI v '(1)=167 SZ v '(1)=084 W'(1)=3304 CI= 1657 CJ S (1)= PI v ' (1) = 31500

17 -585- SZ v (x)= 36x x x x x 565 W(x)=6x x x x x 565 ( x) =6x 6 +6x 7 +6x 8 +6x 9 +3x CJ S(x)=40x + 48x + 56x x x x x 9 PI v (x) = 176x 18 SZ v (x)=40x x x x 4096 W(x)=8x x x x 4096 ( x) = 8x 5 +8x 6 +8x 7 +4x CJ S(x)=48x x x + 7x x + 7x x x x 11 PI v (x) = 80x 00 SZ v (x)=48x x x x x W(x)= 8x x 44 +8x x x ( x) = 8x 6 +8x 7 +8x 8 +8x 9 +4x 10 SZ v '(1)= W'(1)= CI= 440 CJ S'(1)= 58 PI v '(1) =58 SZ v '(1)=45315 W'(1)=6731 CI= 9840 CJ S'(1)= PI v ' (1)= SZ v '(1)= W'(1)=05168 CI= I graphs which ar ot partial cubs, li th cos C(4,), ) o ca us a procdur basd o * which is th trasitiv closur of Wilr s rlatio. 60,61 Numrical calculatio wr do by our origial softwar programs TOPOCLUJ, 3 Omga coutr 6 ad Nao Studio. 63 Th us of th hri discussd dscriptors i corrlatig of various physicochmical or biological proprtis with th molcular structur hav b xtsivly prov, thus w oly ivit th radr to cosult som moographs i th fild. 54,64-67 Coclusios Th most usd topological idics: Wir idx W(G), Szgd idx SZ(G), Cluj idics CJ(G) ad th mor rctly dfid Cluj-Ilmau CI(G), wr drivd hr by dg cuttig procdurs i som bipartit graphs ad/or partial cubs. Th aalytical formulas abld us to fid a clar rlatdss amog ths topological dscriptors. Numrical xampls wr giv. Acowldgmts: Th fiacial support of th Romaia Grat CNCSIS PN-II IDEI 506/007, is acowldgd.

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