CLUJ AND RELATED POLYNOMIALS IN BIPARTITE HYPERCUBE HYPERTUBES

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1 SDIA UBB CHEMIA LXI Tom II 0 p. 8-9 RECOMMENDED CITATION Dedcated to Pofeo Eml Codoș o the occao of h 80 th aeay CLUJ AND RELATED POLYNOMIALS IN BIPARTITE HYPERCUBE HYPERBES MAHBOUBEH SAHELI a AMIR LOGHMAN a MIRCEA V. DIUDEA b* ABSTRACT. A oel cla of coutg polyomal called Cluj polyomal wa popoed o the goud of Cluj matce. The polyomal coeffcet ae calculated fom the aboe matce o by mea of othogoal edge-cut. I th pape Cluj polyomal bpatte hypecube hypetube peeted. Defto ad elato wth othe polyomal ad topologcal dce ae deed. Keywod: Cluj polyomal ete-padmaka-ia de Wee de. INTRODUCTION A fte equece of ome gaph-theoetcal categoe/popete uch a the dtace degee equece o the equece of the umbe of k- depedet edge et ca be decbed by o-called coutg polyomal: k P G p G k k whee pgk the fequecy of occuece of the popety patto of G p G P G of legth k ad mply a paamete to hold k. I the Mathematcal Chemty lteatue the coutg polyomal hae ft bee toduced by Hooya []. Cluj dce ad polyomal hae bee toduced by Dudea [-]. I bpatte gaph the coeffcet of CJ polyomal ca be calculated by a othogoal edge-cut pocedue [7-9]. Fo th a theoetcal backgoud eeded. a Depatmet of Mathematc Payame Noo Uety PO BOX Teha Ia. b Depatmet of Chemty Faculty of Chemty ad Chemcal Egeeg Babe-Bolya Uety 0008 Cluj Romaa. * Coepodg autho: dudea@chem.ubbcluj.o

2 M. SAHELI A. LOGHMAN M. V. DIUDEA A gaph G a patal cube f t embeddable the -cube Q whch the egula gaph whoe etce ae all bay tg of legth two tg beg adjacet f they dffe eactly oe poto. The dtace fucto the -cube the Hammg dtace. A hypecube ca alo be epeed a the Catea poduct: Q K. Fo ay edge e=u of a coected gaph G let u deote the et of etce lyg cloe to u tha to : u wv G d w u d w. It follow that u wv G d w d w u. The et ad ubgaph duced by thee etce u ad u ae called emcube of G; the emcube ae called oppote emcube ad ae djot [0]. A gaph G bpatte f ad oly f fo ay edge of G the oppote emcube defe a patto of G: u u V G. Thee emcube ae jut the ete pomte ee aboe of the edpot of edge e=u whch CJ polyomal cout. I patal cube the emcube ca be etmated by a othogoal edge-cuttg pocedue. The othogoal cut fom a patto of the edge G: EG cc... c cc j. k j To pefom a othogoal edge-cut take a taght le egmet othogoal to the edge e ad teect e ad all t paallel edge a plae gaph. The et of thee teecto called a othogoal cut c k e k=..k ma. A eample ge Fgue. = =9 = =9 CJP = CJP =89090=CJPG=SZ Fgue. Cuttg pocedue the calculato of eeal topologcal decpto 8

3 CLUJ AND RELATED POLYNOMIALS IN BIPARTITE HYPERCUBE HYPERBES To ay othogoal cut c k two umbe ae aocated: ft oe epeet the umbe of edge e k teected o the cuttg cadalty c k whle the ecod k o the umbe of pot lyg to the left had wth epect to c k. Becaue bpatte gaph the oppote emcube defe a patto of etce t ealy to detfy the two emcube: u = k ad u = - k o ce-ea. By th cuttg pocedue thee cae hae to be codeed a ummazed Table. Table : Mathematcal opeato ad defed thee polyomal Opeato Polyomal ame Fomula k k Summato Cluj-Sum Pawe ummato Pawe poduct ete-padmaka-ia Cluj-Poduct k k e k k SZ e e The ft deate fo = of a coutg polyomal pode gle umbe ofte called topologcal dce. It ealy ee that the ft deate = of the ft two polyomal ge oe ad the ame alue but the ecod deate dffeet ad the followg elato hold ay gaph [-] ; I bpatte gaph take the mamal alue amog all the gaph o the ame umbe of etce: e E G V G Th eult ca be ued a a cteo fo the bpatty of a gaph [78]. The thd polyomal CJP ue the pawe poduct; t pecely the ete Szeged polyomal SZ defed by Ahaf et al [-]. Th come out fom the elato betwee the bac Cluj Dudea [] ad Szeged Gutma [] dce: CJP CJDI G SZ G SZ All the thee polyomal ad the deed dce do ot cout the equdtat etce a dea toduced Chemcal Gaph Theoy by Gutma. We call thee polyomal of ete pomty. 87

4 M. SAHELI A. LOGHMAN M. V. DIUDEA LATTICE BUILDING I ome ecet pape [7] Dudea et al. popoed the embeddg of -Cube uface othe tha the phee. I cae of ope tube ome eample ae ge Fgue ; oe lde ; oe lde Fgue. Eample of hypecube embedded the aotube 88

5 CLUJ AND RELATED POLYNOMIALS IN BIPARTITE HYPERCUBE HYPERBES 89 RESULTS AND DISCUSSION All the thee polyomal lted Table ae eemplfed fo ome bpatte hypecube hypetube Table to 9. The aalytcal fomula wee deed by umecal aaly. Numecal calculato hae bee doe by TOPOCLUJ oftwae [8]. Table : Cluj ad elated polyomal Numbe of etce ad edge V E ; ` ad `` Cluj polyomal ad t deate 7 Szeged polyomal ad de 8 Sz 9 Sz

6 M. SAHELI A. LOGHMAN M. V. DIUDEA 90 Table : Cluj ad elated polyomal Numbe of etce ad edge V E ; ` ad `` Cluj polyomal ad t deate 7 8 Szeged polyomal ad de 8 Sz 9 Sz 9 Table : Eample: ; ` ad ``; umbe of etce ad edge Polyomal ` `` e

7 CLUJ AND RELATED POLYNOMIALS IN BIPARTITE HYPERCUBE HYPERBES Table : Eample: Cluj polyomal ad t deate Cluj polyomal CJ` CJ`` Table : Eample: Szeged polyomal ad de Szeged polyomal Szeged de Table 7: Eample: ; ` ad ``; umbe of etce ad edge 8 Polyomal ` `` e

8 M. SAHELI A. LOGHMAN M. V. DIUDEA Table 8: Cluj polyomal ad t deate 8 Cluj polyomal CJ` CJ`` Table 9: Eample; Szeged polyomal ad de Szeged polyomal Szeged de CONCLUSION I th pape we peeted the calculato of Cluj polyomal hypecube hypetube. Defto ad elato wth othe polyomal ad the coepodg topologcal dce wee ge. Aalytcal fomula a well a eample wee tabulated. REFERENCES. H. Hooya Dcete Appl. Math M. V. Dudea J. Chem. If. Comput. Sc M. V. Dudea Coat. Chem. Acta M. V. Dudea A. E. Vztu ad D. Jaežč J. Chem. If. Model

9 CLUJ AND RELATED POLYNOMIALS IN BIPARTITE HYPERCUBE HYPERBES. M. V. Dudea J. Math. Chem M. V. Dudea N. Doot A. Iamaeh Cluj Cj Polyomal ad Idce a Dedtc Molecula Gaph Studa U. "Babe-Bolya" Chema M. V. Dudea Noel Molecula Stuctue Decpto-Theoy ad Applcato I M. V. Dudea Noel Molecula Stuctue Decpto-Theoy ad Applcato II I. Gutma ad S. Klaža J. Chem. If. Comput M. V. Dudea ad S. Klaža Acta. Chem. Sloe P. V. Khadka Nat. Acad. Sc. Lett M. H. Khalfeh H. Youef-Aza ad A. R. Ahaf Lea Algeba Appl A. R. Ahaf M. Ghoba ad M. Jalal J. Theo. Comput. Chem T. Maou ad M. Schok Dc. Appl. Math I. Gutma Gaph Theoy Note A. Paa-Moldoa M. V. Dudea Ia. J. Math. Chem K. Fathalkha A. Pîa-Moldoa M. V. Dudea Studa U. Babe-Bolya Chema O. Uu M. V. Dudea TOPOCLUJ oftwae pogam Babe-Bolya Uety 00. 9