/
Algebraicproperties,IIWewereledtomaketwoconjectures,asfollows.
... Algebraicproperties,IIWewereledtomaketwoconjectures,asfollows.
...

Algebraicproperties,IIWewereledtomaketwoconjectures,asfollows. ... - PDF document

natalia-silvester
natalia-silvester . @natalia-silvester
Follow
366 views
Uploaded On 2016-05-20

Algebraicproperties,IIWewereledtomaketwoconjectures,asfollows. ... - PPT Presentation

Conjecture1Theanconjecture LetabeanalgebraicintegerThenthereexistsanaturalnumbernsuchthatanisachromaticroot Conjecture2Thenaconjecture LetabeachromaticrootThennaisachromaticrootforanynatur ID: 327364

Conjecture1(Thea+nconjecture). Letabeanalgebraicinteger.Thenthereexistsanaturalnumbernsuchthata+nisachromaticroot. Conjecture2(Thenaconjecture). Letabeachro-maticroot.Thennaisachromaticrootforanynatur

Share:

Link:

Embed:

Download Presentation from below link

Download Pdf The PPT/PDF document "Algebraicproperties,IIWewereledtomaketwo..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.


Presentation Transcript

Algebraicproperties,IIWewereledtomaketwoconjectures,asfollows. Conjecture1(Thea+nconjecture). Letabeanalgebraicinteger.Thenthereexistsanaturalnumbernsuchthata+nisachromaticroot. Conjecture2(Thenaconjecture). Letabeachro-maticroot.Thennaisachromaticrootforanynaturalnumbern.Ifthea+nconjectureistrue,wecanask,forgivena,whatisthesmallestnforwhicha+nisachromaticroot?AnexampleThegoldenratioa=(p 5�1)/2isnotachro-maticroot,asitliesin(0,1).Also,a+1anda+2arenotchromaticrootssincetheiralgebraicconjugatesarenegativeorin(0,1).However,therearegraphs(e.g.thetrun-catedicosahedron)whichhavechromaticrootsveryclosetoa+2,theso-called“goldenroot”.Wedonotknowwhethera+3isachromaticrootornot.However,a+4isachromaticroot(thesmallestsuchgraphhaseightvertices),andhencesoisa+nforanynaturalnumbern4.Quadraticroots Theorem3. Letabeanintegerinaquadraticnumbereld.Thenthereisanaturalnumbernsuchthata+nisaquadraticroot.Ifaisirrational,thenthesetfa+n:n2Zgisthesetofallquadraticintegerswithgivendis-criminant.Soitisenoughtoshowthat,foranynon-squaredcongruentto0or1mod4,thereisaquadraticintegerwithdiscriminantdwhichisachromaticroot.Iwillsketchtheideasbehindtheproofofthisandpartialresultsforhigher-degreealgebraicin-tegers. RingsofcliquesAringofcliquesisthegraphR(a1,...,an)whosevertexsetistheunionofn+1completesubgraphsofsizes1,a1,...,an,wheretheverticesofeachcliquearejoinedtothoseofthecliquesimmedi-atelyprecedingorfollowingitmodn+1. Theorem4(Read). ThechromaticpolynomialofR(a1,...,an)isaproductoflinearfactorsandthepoly-nomial1 q nĂ•i=1(q�ai)�nĂ•i=1(�ai)!.Wecallthistheinterestingfactor.Examples  Ifai=1foralli(sothatthegraphisan(n+1)-cycle),theinterestingfactoris((q�1)n�(�1)n)/q=(xn�(�1)n)/(x+1),wherex=q�1.Itsrootsare2nthrootsofunity.Inpar-ticular,ifnisprime,thisfactorisirreducibleanditsGaloisgroupiscyclicofordern�1.  Ifn=3,theinterestingfactorofR(1,1,4)isq2�7q+11,withroots(7p 5)/2.Thisistheeight-vertexgraphpromisedearlier.QuadraticandcubicintegersForn=3,theinterestingfactorofR(a,b,c)isx2�(a+b+c)x+(ab+bc+ca).Thediscriminantofthisquadraticis(a+b+c)2�4(ab+bc+ca).Ittakesbutalittleingenuitytoshowthatthisdiscriminanttakesallpossiblevaluescongruentto0or1mod4.Forn=4,wehaveafour-parameterfamilyofcubicsfortheinterestingfactors.Aretheseenoughtoprovethea+nconjectureforcubicinte-gers?(Wehavealonglistofcubicsobtainedfromthisconstructionbutdon'tseemtohavehitevery-thing!)Ahigher-dimensionalfamilyLetGbeagraphwhosevertexsetistheunionoftwocliques,ofsizesnandm.Fori=1,...,m,letFibethesetofneighboursintherstcliqueoftheithvertexofthesecond.WemayassumewithoutlossofgeneralitythattheunionofallthesetsFi 2 OtherfamiliesofgraphsWehavedonesimilaranalysisonotherfamiliesofgraphs,including  completebipartitegraphs;  “theta-graphs”(oneoftheseconsistsofppathsoflengthswiththeendpointsidenti-ed)–thesewerethegraphsusedbySokaltoshowthatchromaticrootsaredenseinthecomplexplane;  smallgraphs.Theresultsaresimilarbutthereisnotimetopresentthemhere.FurtherspeculationTheGaloisgroupofa“random”polynomialistypicallythesymmetricgroupofitsdegree.Thechromaticpolynomialofarandomgraphcannotbeirreducible,sinceitwillhavemanylin-earfactorsq�m,formuptothechromaticnum-ber.Bollob´asshowedthatthechromaticnumberisalmostsurelycloseton/(2log2n). Wildspeculation5. Thechromaticpolynomialofarandomgraphisalmostsurelyaproductoflinearfac-torsandoneirreduciblefactorwhoseGaloisgroupisthesymmetricgroupofitsdegree. 4