On the Multiplicative Zagreb Indices of Bucket Recursive Trees

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Willa Houston
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1 Iraa J a Cem 8 arc Iraa Joural of aemacal Cemsry Joural omepage: wwwmcasauacr O e ulplcae agre Ices of Buce Recurse Trees RAIN KAI Deparme of Sascs Imam Kome Ieraoal ersy Qaz Ira ARTICL INFO Arcle Hsory: Recee Feruary 05 Accepe Aprl 05 ulse ole 5 Sepemer 06 Acaemc or: Al Reza Asraf Keywors: Buce recurse rees ulplcae agre e ome geerag fuco omes ABSTRACT Buce recurse rees are a eresg a aural geeralzao of orary recurse rees a ae a coeco o maemacal cemsry I s paper we ge e lower a upper ous for e mome geerag fuco a momes of e mulplcae agre ces a raomly cose uce recurse ree of sze w mamal uce sze Also we coser e rao of e mulplcae agre ces for ffere alues of a All our resuls reuce o e orary recurse rees for 07 ersy of Kasa ress All rgs resere INTRODCTION Trees are efe as coece graps wou cycles Recurse rees are rooe laelle rees were e roo s laelle y a e laels of all successors of ay oe are larger a e lael of [8] I s of parcular eres applcaos o assume e raom recurse ree moel a o spea aou a raom recurse ree w oes wc meas a oe of e! possle recurse rees w oes s cose w equal proaly e e proaly a a parcular ree w oes s cose s always /! A eresg a aural geeralzao of raom recurse rees as ee rouce [7] a ese are calle uce recurse rees I s moel e oes of a uce recurse ree are uces wc ca coa up o a fe eger amou of laels A proalsc escrpo of raom uce recurse rees s ge y a geeralzao of e socasc grow rule for orary raom recurse rees wc s e specal sace I fac a ree grows y progresse araco of creasg eger laels: we serg lael o a esg uce recurse ree coag Correspog Auor: mal aress: razem@sciuacr DOI: 005/mc075385

2 38 KAI laels e coag e laels { } all esg laels e ree compee o arac e lael were all esg laels ae equal cace o recru e ew lael If e lael wg s compeo s coae a oe w less a laels a usaurae uce lael s ae o s oe oerwse f e wg lael s coae a oe w laels alreay a saurae uce lael s aace o s oe as a ew uce coag oly e lael Sarg w a sgle uce as e roo oe coag oly e lael afer sero seps were e laels 3 are successely sere accorg o s grow rule resuls a so calle raom uce recurse ree w laels a mamal uce sze For a esg uce recurse ree T w laels e proaly a a cera oe T w capacy c aracs e ew lael s equal o e umer of laels coae e c/ see [7] Fgure llusraes a uce recurse ree of sze w mamal uce sze For a coeco o cemsry suppose aoms a ermer a repeely race molecule are socascally laelle w egers e laelle aoms a fucoal group ca e cosere as e laels of a uce a uce recurse ree I s oous a e umer of oes ere uces a uce recurse ree T s less a for > Tus we ca sow e sze of e ree as a fuco of a Le e a real alue fuco of were 0 a for all Now we ca wre e sze of e ree as e V T We coose e fuco s form for relao ewee e uce recurse rees a orary recurse rees Fgure : A uce recurse ree of sze w mamal uce sze [6] Two erces of grap G coece y a ege are sa o e aace Te umer of erces of G aace o a ge ere s e egree of s ere a wll e eoe y Toesc e al [9 0] ae suggese o coser mulplcae aras of ae grap aras wc apple o e agre ces

3 O e ulplcae agre Ices of Buce Recurse Trees 39 woul lea o e mulplcae agre ces of a grap G eoe y a G uer e ame frs a seco mulplcae agre e respecely Tese are efe as G a G V G G u u G were V G a G are e ere se a ege se of G respecely [3] I proaly eory a sascs e mome geerag fuco of a raom arale s a alerae specfcao of s proaly sruo Tus proes e ass of a alerae roue o aalycal resuls compare w worg recly w proaly esy fucos or cumulae sruo fucos Tere are parcularly smple resuls for e mome geerag fucos of sruos efe y e wege sums of raom arales Noe oweer a o all raom arales ae mome geerag fucos Defo Te mome geerag fuco of a raom arale X s efe as X ep X R wereer s epecao ess Te reaso for efg s fuco s a ca e use o f all e momes of e sruo I fac X 0! were s e mome of X e X [] RSLTS Le eoe e egree of uce our moel of sze w mamal uce sze a e e frs mulplcae agre e We also efe o e e sgma fel geerae y e frs sages [] If lael s aace o a usaurae uce e Bu f lael s aace o a saurae uce e y e socasc grow rule of e ree a y efo of e frs mulplcae agre e

4 0 KAI 3 were s uformly srue o uces se Teorem Le ep e e mome geerag fuco of of a uce recurse ree of sze w mamal uce sze Te ep roof We ae! 0 were s e mome of For T V c sce s -measurale a e lael s aace o ay saurae uce of e alreay grow ree T w proaly c Tus Tag epecao of e equaly : 5 Also Tus 5 leas o 6 a proof s complee

5 O e ulplcae agre Ices of Buce Recurse Trees If we replace y l e we oa e upper ou for e proaly geerag fuco [] Le e e seco mulplcae agre e of a uce recurse ree of sze w mamal uce sze Te y efo of e seco mulplcae agre e 7 Teorem Le ep N e e mome geerag fuco of of a uce recurse ree of sze w mamal uce sze Te ep N roof Le e e mome of of a uce recurse ree of sze w mamal uce sze For smlar o e frs mulplcae agre e c Tus 8 Tag epecao of e equaly 8: Now proof s complee us smlar o e proof of Teorem I passg we coser e rao of e mulplcae agre ces for ffere alues of a

6 KAI Teorem 3 Suppose {} a Te a roof We ae Le g for 0 > Te g s coe ecause 0 3 g a y Jese s equaly X X Tus W e same maer we ca oa e upper ou for Teorem Suppose a

7 O e ulplcae agre Ices of Buce Recurse Trees 3 S K Te K a S roof By efo of e cooal epecao K W e same maer we ca oa e lower ou for S Corollary 5 Te presee resuls Teorem reuce o e preous resuls Teorem for Teorem 6 Suppose a < Te a roof From

8 KAI W e same maer we ca oa e lower ou of We ca suy e rao of e mulplcae agre ces for ffere alues of as a are ffere w e aoe presee approac Corollary 7 For orary recurse rees ep N ep Also le r [ ] w / r / By Holer s equaly Also a r m r m 6 6 Te e ous oes o epe o a orary recurse rees 3 DISCSSION AND CONCLSION So far e mulplcae agre ces ae ee sue asly leraure from maemacal po of ew I s paper we rouce e frs proalsc aalyss of e mulplcae agre ces e raom uce recurse rees Troug e recurrece equaos a upper ou relae o e frs mulplcae agre e a a lower ou relae o e seco mulplcae agre e are oae As a eresg resul s sow a ese ous are e same s moel I s ffcul o

9 O e ulplcae agre Ices of Buce Recurse Trees 5 f a lower ou Teorem a a upper ou Teorem sce e mamum egree of uces of our moel mg o cage for ffere alues of Howeer we ca suy some proalsc caracerscs of ese ces suc as margales asympoc ormaly a so o see [ 5 6] for eals Te lower a upper ous for e mome geerag fuco a momes are ery mpora For eample y aro s equaly 9 0 las e al [] cosere a mulplcae erso of e frs agre e efe as G u u G W e same approac we ca oa e lower a upper ous relae o s e Geerally oe ca ee s approac o aoer ces a ree srucures RFRNCS Bllgsley roaly a easure Jo Wley a Sos New Yor 995 las A Iramaes I Guma ulplcae ersos of frs agre e ATCH Commu a Compu Cem A Iramaes A Hossezae I Guma O mulplcae agre ces of graps Iraa J a Cem R Kazem roalsc aalyss of e frs agre e Tras Com R Kazem Dep uce recurse rees w arale capaces of uces Aca a S gl Ser R Kazem Te eccerc coecy e of uce recurse rees Iraa J a Cem H amou R T Smye roalsc aalyss of uce recurse rees Teore Compu Sc A er J W oo O e alue of oes raom rees Caaa J a R Toesc D Ballao V Coso Noel molecular escrpors ase o fucos of ew ere egrees : I Guma B Furula s Noel olecular Srucure Descrpors Teory a Applcaos I Kragueac Kragueac R Toesc V Coso New local ere aras a molecular escrpors ase o fucos of e ere egrees ATCH Commu a Compu Cem

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