The Complete Graph: Eigenvalues, Trigonometrical Unit-Equations with associated t-complete-eigen Sequences, Ratios, Sums and Diagrams
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1 The Complee Gph: Eigevlues Tigoomeicl Ui-Equios wih ssocied -Complee-Eige Sequeces Rios Sums d Digms Pul ugus Wie* Col Lye Jessop dfdeemi Je dewusi bsc The complee gph is ofe used o veify cei gph heoeicl defiiios d pplicios Regdig he dcecy mix ssocied wih he complee gph s cicul mix we fid is eigevlues d use his esul o geee igoomeicl ui-equios ivolvig he sum of ems of he fom [ / ] whee is odd This gives ise o-complee-eige sequeces ddigms simil o he fmous Fey sequece d digm We showh he io ivolvig sum of he ems of he-complee eige sequece coveges o ½ d use his io o fid he -complee eige e To fid he eigevlues ssocied wih he chceisic polyomil of complee gph usig iducio we cee geel deemi equio ivolvig he mio of he mix ssocied wih his chceisic polyomil ey wods: complee gph igoomeicl equios eigevlues sequeces MS clssificio:0c0 *Coespodig uho: Mhemics Dub Souh fic 0 emil:wiep@uzcz ORCD D N-0
2 oducio We use he gph-heoeicl oio of is e l Ofe whe ew gph-heoeicl defiiio is ioduced he defiiio is esed o he complee gph Fo exmple complee gph o veices hs miimum veex coveig isig of y se of - veices The umbe of spig ees is well ow so is is chomic umbe dius d dimee ec The eigevlues of he dcecy mix ssocied wih he complee gph is lso esy o compue see Bouwe d emes fo exmple They e - d - Cosideig he dcecy mix of he complee gph s cicul mix we fid is eigevlues i ems of sie d ie Usig he ie p d odd d he fc h - is eigevlue we geeeigoomeicl-ui equios: These equio esuled i -complee-eige sequeces d usig ui mio pis d digms simil o h of he fmous Fey sequece d digm We show h he io ivolvig he sum of he ems of he -complee eige sequece coveges o ½ d evlue e usig his io Thee e my ow mehods vilble o fid he eigevlues ssocied wih he complee gph see Jessop Some mehods e sho ohes e log bu mhemiclly ieesig lhough he iducio mehod c be egded s lboious i illuses he viey of cei combioil specs ssocied wih he deemis ivolved wih he chceisic polyomil ssocied wih he mix of he complee gph which we demose i he heoem i secio below Eigevlues of he complee gph fom cicul mix-eige sequeces Cosideig he mix of he complee gph s cicul mix we fid is eigevlues o cee igoomeicl ui-equios The esuls of he followig Lemms c be foud i Jessop
3 Lemm Le be x cicul mix The he eigevecos of he cicul mix e give by: 0 v T whee i exp e he h oos of uiy d i The coespodig eigevlues e he give by 0 0 Lemm Le be he dcecy mix of he complee gph o veices The x d is eigevlues e fo ll whee 0 i i i e e e i i i si si si si i Usig he bove Lemm d he fc h he eigevlues of he dcecy mix ssocied wih he complee gph e oce d mulipliciy we hve he followig heoem:
4 Theoem Poof Fo 0 he bove lemm yields he eigevlue So fo 0 Thus fo 0 he eigevlues e isi Now fo 0 we ide si si si si si si si si B si whee hs he fis ems d B he ex ems ddig he fis em of d he ls em of B yield: si si si si si si 0 Geelly ddig he h em of d he si si h em of B whee yields
5 0 si si si si Theefoefo 0 0 si d he si i Now B hs he fis ems d B he ex ems ddig he fis em of d he ls em of B yield: The -h em of d he fis em of B yield:
6 6 ddig he secod em of d he secod o ls em of B: Geelly ddig he -h em of d he -h em of B Thus which yields which yields We heefoe geee he followig igoomeicl ui-equios hvig ems ivolvig whee ivolves ll odd iol umbes i he ievl 0 ie is lso oddthee will be excly such odd iol umbes fomig -sequece:
7 is he oly odd iol umbe bewee 0 d e he odd iol umbes bewee 0 d e he hee odd iol umbes bewee 0 d e he ems of he sequece Fo ech we heefoe ssocie he -sequeceof odd iol ems ech em belogig o he ievl 0 d hvig he fom odd coiig ems: This sequece hs similiies o he Fey sequece The Fey sequece of ode is he sequece FY of compleely educed fcios bewee 0 d which whe i lowes ems hve deomios less h o equl o ged i ode of icesig size seedy dwigh Fey sequeces e med fe he BiishgeologisJoh Fey S whose lee bou hese sequeces ws published i he Philosophicl Mgzie i 86 The sequece we deived fom usig he eigevlues of he complee gph is clled he -complee-eige sequece Coolly The sum of he ems of he -complee eige sequece: is give by: Poof Wiig ech -sequece dow wice wih he secod evesed we ge:
8 8 6 ddig coespodig ems we ge double he sum of he ems of he sequece: Theefoe which gives he esul f we fom he io of he -complee-eige sequeceby dividig ech em of he oigil - complee sequece by we obi he sequece d which coveges o he vlue of s iceses So is he -complee-eigeio of o which coveges o he vlue of This gives he followig coolly: Coolly lim lim
9 9 Fo he sequece S ssocie he mio imge ui-pi pe belogig o he ui-mio -complee eige sequece: 6 ' S of he fom c whee c is eve The sum of coespodig pis of ems fom S d ' S yields Thus e ui-mio pis The uio of S d ' S yields he ol -complee eige sequece: 6 ' S S d Joiig eighbos d ui mio pis we cee he digm fo simil o he Fey sequece digm: / / / / / 6/ Figue : Digm fo he ol -complee eige sequece fo The vege degee of he veices of he complee gph o veices is
10 0 chig he vege degee of he complee gph o veices o he iegl of he-complee-eigeio wih espec o we fom he -complee eige esee Wie d dewusi d Wie d Jessop: d d d d l c The fis -complee eigesequece ises whe d The sequece is is: So c l so h he -complee eige e d l c 0 l l
11 ducio d he eigevlues of he complee gph Thee e my diffee mehods vilble o fid he eigevlues ssocied wih he complee gph see Jessop Some mehods e sho ohes e log bu eleg lhough he iducio mehod is log i illuses he iiguig specs ssocied wih he deemis ivolved wih chceisic polyomil ssocied wih he mix of he complee gph which we demose i he heoem below The followig heoem is used i he poof of fidig he eigevlues of he complee gph ivolves he deemi of mio of mix whee is he dcecy mix of he complee gph o veices Theoem f x whee is x mix wih he x de de
12 Poof by iducio Fo de de de de de de de de de de de de de de de de de de
13 de de de de de de de de de de ssume he hypohesis i ue fo ll ie de fo ll The fo de de x The expdig log he fis ow
14 de x x de de The fis em is obied fom he expsio of he fis colum i he fis ow d he secod ems isobied fom he ideicl ems obied fom he expsio of he d o h colums Now x de de d x The de de de de de de de de de de de de de de de de de de de de de de de de
15 Now he ledig mus hve powe so h we ge de d de which e boh ow So coiuig de de de de de de de de de Subsiuig de d de fo ll we ge de Fcoisig ou of he ems i he sque bces we ge de Woig wih he fis wo ems i sque bces we ge de Tig ou he ex fco of fom iside he sque bces we ge de
16 6 6 Woig wih he fis wo ems i sque bces we ge: de Noe h he fis em i he sque bces compises of We do he sep bove ol of imes ig ou he fco o ge de Noe h he powe of i he fis em i he sque bces is d he powe of i he secod em is lso Simplifyig we ge de This cocludes he poof by iducio h de fo ll
17 Coolly Le be he dcecy mix of he complee gph o veices The x hs eigevlue wih mulipliciy d eigevlue - wih mulipliciy ece de Poof of Coolly by iducio Fo 0 0 de de Noe h he eigevlues of e λ = - ime d λ= oce ssume he hypohesis i ue fo ie de de x ie fo imes d The fo oce
18 8 de x x de de de de Now pplyig he iducive hypohesis fo de d Theoem fo de we ge de ie imes d oce So we hve poved h he eigevlues of he dcecy mix of he complee gph e d d h he chceisic polyomil is P The wo fcos d give ise o he qudic which hs he ssocied couge pis
19 9 Coclusio Regdig he dcecy mix ssocied wih he complee gph s cicul mix we fomedhe ui-equios: Fo ech we heefoe geeed he -sequece of odd iol ems ech i he ievl 0 d hvig he fom : This sequece is efeed o s he -complee-eige sequece d we showed h he sum of is ems is d h he io of his sum o coveges o We use he ssocied ol -complee eige sequece o uc he digm ivolvig ui mio pis d foud he -complee eige e by usig iegio combied wih he vege degee of he complee gph o veices o be: l l ode o fid he eigevlues of he dcecy mix ssocied wih he complee gph by iducio we geeed equio ivolvig he deemi of he mio of he mix ssocied wih he chceisic polyomil of his dcecy mix Refeeces Bouwe E emes 0 W Spec of GphsSpige New Yo is J M is J L d Mossighoff M 008 Combioics d Gph heoyspige New Yo dy G Wigh EM 99 oducio o he Theoy of Numbes Fifh EdiioOxfod Uivesiy Pess
20 0 Jessop C L Mices of Gphs d Desigs wih Emphsis o hei Eige-Pi Blced Chceisic 0 M Sc Disseio Uivesiy of w-zulu Nl Wie P d dewusi FJ 0Tee-cove io of gphs wih sympoic covegece ideicl o he secey poblemdvces i Mhemics: Scieific Joul : -6 Wie P d Jessop CL 0egl eige-pi blced clsses of gphs wih hei io sympoe e d ivoluio complemey specseiol Joul of Gph Theoy icle D pges
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