Spectral generalizations of line graphs: on graphs with least negative eigenvalue -2 PDF
Preview Spectral generalizations of line graphs: on graphs with least negative eigenvalue -2
LONDONMATHEMATICALSOCIETYLECTURENOTESERIES ManagingEditor:ProfessorN.J.Hitchin,MathematicalInstitute, UniversityofOxford,24–29StGiles,OxfordOX13LB,UnitedKingdom Thetitlesbelowareavailablefrombooksellers,orfromCambridgeUniversityPressatwww.cambridge.org 100 Stoppingtimetechniquesforanalystsandprobabilists, L.EGGHE 105 Alocalspectraltheoryforclosedoperators, I.ERDELYI&WANGSHENGWANG 107 CompactificationofSiegelmodulischemes, C.-L.CHAI 109 Diophantineanalysis, J.LOXTON&A.VANDERPOORTEN(eds) 113 Lecturesontheasymptotictheoryofideals, D.REES 116 Representationsofalgebras, P.J.WEBB(ed) 119 Triangulatedcategoriesintherepresentationtheoryoffinite-dimensionalalgebras, D.HAPPEL 121 ProceedingsofGroups–StAndrews1985, E.ROBERTSON&C.CAMPBELL(eds) 128 Descriptivesettheoryandthestructureofsetsofuniqueness, A.S.KECHRIS&A.LOUVEAU 130 Modeltheoryandmodules, M.PREST 131 Algebraic,extremal&metriccombinatorics, M.-M.DEZA,P.FRANKL&I.G.ROSENBERG(eds) 141 Surveysincombinatorics1989, J.SIEMONS(ed) 144 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Ergodictheoryanditsconnectionswithharmonicanalysis, K.PETERSEN&I.SALAMA(eds) 207 GroupsofLietypeandtheirgeometries, W.M.KANTOR&L.DIMARTINO(eds) 208 Vectorbundlesinalgebraicgeometry, N.J.HITCHIN,P.NEWSTEAD&W.M.OXBURY(eds) 209 Arithmeticofdiagonalhypersurfacesoverinfinitefields, F.Q.GOUVE´A&N.YUI 210 HilbertC∗-modules, E.C.LANCE 211 Groups93Galway/StAndrewsI, C.M.CAMPBELLetal(eds) 212 Groups93Galway/StAndrewsII, C.M.CAMPBELLetal(eds) 214 GeneralisedEuler-Jacobiinversionformulaandasymptoticsbeyondallorders, V.KOWALENKOetal 215 Numbertheory1992–93, S.DAVID(ed) 216 Stochasticpartialdifferentialequations, A.ETHERIDGE(ed) 217 Quadraticformswithapplicationstoalgebraicgeometryandtopology, A.PFISTER 218 Surveysincombinatorics,1995, PETERROWLINSON(ed) 220 Algebraicsettheory, A.JOYAL&I.MOERDIJK 221 Harmonicapproximation, S.J.GARDINER 222 Advancesinlinearlogic, J.-Y.GIRARD,Y.LAFONT&L.REGNIER(eds) 223 Analyticsemigroupsandsemilinearinitialboundaryvalueproblems, KAZUAKITAIRA 224 Computability,enumerability,unsolvability, S.B.COOPER,T.A.SLAMAN&S.S.WAINER(eds) 225 Amathematicalintroductiontostringtheory, S.ALBEVERIOetal 226 Novikovconjectures,indextheoremsandrigidityI, S.FERRY,A.RANICKI&J.ROSENBERG(eds) 227 Novikovconjectures,indextheoremsandrigidityII, S.FERRY,A.RANICKI&J.ROSENBERG(eds) 228 ErgodictheoryofZdactions, M.POLLICOTT&K.SCHMIDT(eds) 229 Ergodicityforinfinitedimensionalsystems, G.DAPRATO&J.ZABCZYK 230 Prolegomenatoamiddlebrowarithmeticofcurvesofgenus2, J.W.S.CASSELS&E.V.FLYNN 231 Semigrouptheoryanditsapplications, K.H.HOFMANN&M.W.MISLOVE(eds) 232 ThedescriptivesettheoryofPolishgroupactions, H.BECKER&A.S.KECHRIS 233 Finitefieldsandapplications, S.COHEN&H.NIEDERREITER(eds) 234 Introductiontosubfactors, V.JONES&V.S.SUNDER 235 Numbertheory1993–94, S.DAVID(ed) 236 TheJamesforest, H.FETTER&B.GAMBOADEBUEN 237 Sievemethods,exponentialsums,andtheirapplicationsinnumbertheory, G.R.H.GREAVESetal 238 Representationtheoryandalgebraicgeometry, 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257 Anintroductiontononcommutativedifferentialgeometryanditsphysicalapplications2ed, J.MADORE 258 Setsandproofs, S.B.COOPER&J.TRUSS(eds) 259 Modelsandcomputability, S.B.COOPER&J.TRUSS(eds) 260 GroupsStAndrews1997inBath,I, C.M.CAMPBELLetal 261 GroupsStAndrews1997inBath,II, C.M.CAMPBELLetal 262 Analysisandlogic, C.W.HENSON,J.IOVINO,A.S.KECHRIS&E.ODELL 263 Singularitytheory, BILLBRUCE&DAVIDMOND(eds) 264 Newtrendsinalgebraicgeometry, K.HULEK,F.CATANESE,C.PETERS&M.REID(eds) 265 Ellipticcurvesincryptography, I.BLAKE,G.SEROUSSI&N.SMART 267 Surveysincombinatorics,1999, J.D.LAMB&D.A.PREECE(eds) 268 Spectralasymptoticsinthesemi-classicallimit, M.DIMASSI&J.SJO¨STRAND 269 Ergodictheoryandtopologicaldynamics, M.B.BEKKA&M.MAYER 270 AnalysisonLiegroups, N.T.VAROPOULOS&S.MUSTAPHA 271 Singularperturbationsofdifferentialoperators, S.ALBEVERIO&P.KURASOV 272 Charactertheoryfortheoddordertheorem, T.PETERFALVI 273 Spectraltheoryandgeometry, E.B.DAVIES&Y.SAFAROV(eds) 274 TheMandlebrotset,themeandvariations, TANLEI(ed) 275 Descriptivesettheoryanddynamicalsystems, M.FOREMANetal 276 Singularitiesofplanecurves, E.CASAS-ALVERO 277 Computationalandgeometricaspectsofmodernalgebra, M.D.ATKINSONetal 278 Globalattractorsinabstractparabolicproblems, J.W.CHOLEWA&T.DLOTKO 279 Topicsinsymbolicdynamicsandapplications, F.BLANCHARD,A.MAASS&A.NOGUEIRA(eds) 280 CharactersandautomorphismgroupsofcompactRiemannsurfaces, THOMASBREUER 281 Explicitbirationalgeometryof3-folds, ALESSIOCORTI&MILESREID(eds) 282 Auslander-Buchweitzapproximationsofequivariantmodules, M.HASHIMOTO 283 Nonlinearelasticity, Y.FU&R.OGDEN(eds) 284 Foundationsofcomputationalmathematics, R.DEVORE,A.ISERLES&E.SU¨LI(eds) 285 Rationalpointsoncurvesoverfinitefields, H.NIEDERREITER&C.XING 286 Cliffordalgebrasandspinors2ed, P.LOUNESTO 287 TopicsonRiemannsurfacesandFuchsiangroups, E.BUJALANCE,A.F.COSTA&E.MART`INEZ(eds) 288 Surveysincombinatorics,2001, J.HIRSCHFELD(ed) 289 AspectsofSobolev-typeinequalities, L.SALOFF-COSTE 290 QuantumgroupsandLietheory, A.PRESSLEY(ed) 291 Titsbuildingsandthemodeltheoryofgroups, K.TENT(ed) 292 Aquantumgroupsprimer, S.MAJID 293 SecondorderpartialdifferentialequationsinHilbertspaces, G.DAPRATO&J.ZABCZYK 294 Introductiontothetheoryofoperatorspaces, G.PISIER LondonMathematicalSocietyLectureNoteSeries.314 Spectral Generalizations of Line Graphs On graphs with least eigenvalue −2 Dragosˇ Cvetkovic´ UniversityofBelgrade PeterRowlinson UniversityofStirling SlobodanSimic´ UniversityofBelgrade publishedbythepresssyndicateoftheuniversityofcambridge ThePittBuilding,TrumpingtonStreet,Cambridge,UnitedKingdom cambridgeuniversitypress TheEdinburghBuilding,CambridgeCB22RU,UK 40West20thStreet,NewYork,NY10011-4211,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia RuizdeAlarco´n13,28014Madrid,Spain DockHouse,TheWaterfront,CapeTown8001,SouthAfrica http://www.cambridge.org (cid:4)C DragosˇCvetkovic´,PeterRowlinson,SlobodanSimic´2004 Thisbookisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithout thewrittenpermissionofCambridgeUniversityPress. Firstpublished2004 PrintedintheUnitedKingdomattheUniversityPress,Cambridge TypefaceTimes10/13pt. SystemLATEX2ε [TB] AcatalogrecordforthisbookisavailablefromtheBritishLibrary. LibraryofCongressCataloginginPublicationData Cvetkovic´,DragosˇM. Spectralgeneralizationsoflinegraphs:ongraphswithleastnegativeeigenvalue−2/ DragosˇCvetkovic´,PeterRowlinson,SlobodanSimic´. p. cm. Includesbibliographicalreferencesandindex. ISBN0-521-83663-8(pbk.) 1.Graphtheory. 2.Eigenvalues. I.Rowlinson,Peter. II.Simic,Slobodan III.Title. QA166.C837 2004 511(cid:5).5–dc22 2003065393 ISBN0521836638paperback Inmemoryofourlateparents: JelkaCvetkovic´ (1904–1993) MladenCvetkovic´ (1901–1979) IreneRowlinson (1910–2001) ArthurRowlinson (1908–1996) OlgaSimic´ (1916–2002) KostaSimic´ (1907–1978) v Contents Preface pagexi 1 Introduction 1 1.1 Basicnotionsandresults 1 1.2 Somegeneraltheoremsfromspectralgraphtheory 10 1.3 Elementaryspectralcharacterizations 21 1.4 Ahistoryofresearchongraphswithleasteigenvalue−2 22 2 Forbiddensubgraphs 25 2.1 Linegraphs 25 2.2 Theeigenspaceof−2forgeneralizedlinegraphs 28 2.3 Generalizedlinegraphs 31 2.4 Someotherclassesofgraphs 44 2.5 Generalcharacterizations 51 2.6 Spectralcharacterizationsofregularlinegraphs 54 3 Rootsystems 64 3.1 Grammatricesandsystemsoflines 64 3.2 Somepropertiesof E 69 8 3.3 Extensionsoflinesystems 72 3.4 Smithgraphsandlinesystems 77 3.5 Analternativeapproach 81 3.6 Generalcharacterizationtheorems 83 3.7 CommentsonsomeresultsfromChapter2 86 4 Regulargraphs 88 4.1 Regularexceptionalgraphs 88 4.2 Characterizingregularlinegraphsbytheirspectra 91 vii viii Contents 4.3 Specialcharacterizationtheorems 96 4.4 Regularexceptionalgraphs:computerinvestigations 100 5 Starcomplements 112 5.1 Basicproperties 112 5.2 Graphfoundations 124 5.3 Exceptionalgraphs 129 5.4 Characterizations 131 5.5 Switching 136 6 Themaximalexceptionalgraphs 139 6.1 Thecomputersearch 139 6.2 Representationsin E 142 8 6.3 Aversatilestarcomplement 146 6.4 Graphswithmaximaldegreelessthan28 149 6.5 Thelastsubcase 153 6.6 Concludingremarks 161 7 Miscellaneousresults 164 7.1 Graphswithsecondlargesteigenvaluenotexceeding1 164 7.2 Graphssharingpropertieswiththeircomplements 170 7.3 Spectrallyboundedgraphs 176 7.4 Embeddingagraphinaregulargraph 178 7.5 Reconstructingthecharacteristicpolynomial 180 7.6 Integralgraphs 183 7.7 Graphequations 188 7.8 Othertopics 190 Appendix 193 TableA1 Somegraphsrelatedtographswithleasteigenvalue−2 194 TableA2 Theexceptionalgraphswithleasteigenvaluegreaterthan−2 198 TableA3 Regularexceptionalgraphsandtheirspectra 213 TableA4 Aconstructionofthe68connectedregulargraphswhichare notlinegraphsbutcospectralwithlinegraphs 228 TableA5 One-vertexextensionsofexceptionalstarcomplements 243 TableA6 Themaximalexceptionalgraphs 249 TableA7 The index and vertex degrees of the maximal exceptional graphs 273 Bibliography 281 Indexofsymbolsandterms 295 Preface Theeigenvaluesdiscussedinthisbookarethoseofa(0,1)-adjacencymatrixof afiniteundirectedgraph.Linegraphs,familiartograph-theoristsfordecades, have the property that their least eigenvalue is greater than or equal to −2. Thispropertyissharedwithgeneralizedlinegraphs,whichcanbeviewedas linegraphsofcertainmultigraphs.Apartfromtheseclassesofexamplesthere areonlyfinitelymanyfurtherconnectedgraphswithspectrumintheinterval [−2,∞), and these are called exceptional graphs. This book deals with line graphs,generalizedlinegraphsandexceptionalgraphs,inthecontextofspectral propertiesofgraphs.Havingworkedinspectralgraphtheoryformanyyears,the authorscametoseetheneedforasinglesourceofinformationontheprincipal resultsinthisarea.Workbeganearlyin2000,andtheprincipalmotivationfor writingthebookatthisjuncturewastheconstructionofthemaximalexceptional graphsin1999.Theworkingtitlehasbecomethesubtitleonthegroundsthat ‘Graphswithleasteigenvalue−2’mightappearunreasonablyspecializedtothe casualobserver.Infact,thesubtitleisnotwhollyaccurateinthatitisnecessary totreatalsothegraphswithleasteigenvaluegreaterthan−2. The requirement that the spectrum of a graph lies in [−2,∞) is a natural one, and in principle not a restriction at all. The reason is to be found in the classicalresultofH.Whitney,whoshowedin1932thattwoconnectedgraphs (withmorethanthreevertices)areisomorphicifandonlyiftheirlinegraphs areisomorphic. ThetitlesofChapters2,3and5,namely‘Forbiddensubgraphs’,‘Rootsys- tems’and‘Starcomplements’reflectthreemajortechniquesandthreeperiods inthestudyofgraphswithleasteigenvalue−2.Ofcourse,earlyresultswereof- tenimprovedusinglatertechniques,butonconsideringtheinterplaybetween techniques, the authors decided that a presentation broadly in chronological orderwasthemostnaturalapproach. ix
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