Table Of ContentSpringerBriefs in Mathematics
Carlos Hoppen · David P. Jacobs · Vilmar Trevisan
Locating Eigenvalues
in Graphs
Algorithms and
Applications
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Carlos Hoppen • David P. Jacobs • Vilmar Trevisan
Locating Eigenvalues
in Graphs
Algorithms and Applications
CarlosHoppen DavidP.Jacobs
InstitutodeMatemáticaeEstatística SchoolofComputing
UniversidadeFederaldoRioGrandedoSul ClemsonUniversity
PortoAlegre,RioGrandedoSul,Brazil Clemson,SC,USA
VilmarTrevisan
InstitutodeMatemáticaeEstatística
UniversidadeFederaldoRioGrandedoSul
PortoAlegre,RioGrandedoSul,Brazil
This work was supported by Conselho Nacional de Desenvolvimento Científico e
Tecnológico
ISSN2191-8198 ISSN2191-8201 (electronic)
SpringerBriefsinMathematics
ISBN978-3-031-11697-1 ISBN978-3-031-11698-8 (eBook)
https://doi.org/10.1007/978-3-031-11698-8
MathematicsSubjectClassification:15A18,05C85,05C05,68W40,68P05
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ÀProfa.NairAbreu,cujoentusiasmoe
generosidadenostrouxeàTeoriaEspectral
dosGrafos.
AoProf.UnderwoodDudley,pornosrevelar
abelezadasdemonstrações.
V.Trevisantambémdedicaestelivroà
Eliana,suacompanheiradevida.
Preface
Spectralgraphtheoryisaresearchareathatcombinestoolsandconceptsoflinear
algebra and combinatorics. Its main goal is to determine properties of a graph
throughtheeigenvaluesandeigenvectorsofmatricesassociatedwithit.
Perhaps surprisingly, eigenvalues and eigenvectors turn out to be intimately
connectedwiththestructureofagraph.Intermsofapplications,theyhaveproved
tobeusefulforisomorphismtestingandembeddinggraphsintheplane,forgraph
partitioningandclustering,astopological descriptorsfornetworks andmolecules,
inthegeometricdescriptionofdatasets,andinthedesignofefficientnetworks,just
tomentionafew.Inapurelymathematicalperspective,thestudyofgraphspectra
hasledtoamyriadofopenproblems,rangingfromtheconstructionofgraphswith
a given set of eigenvalues to extremal problems that ask for a characterization of
graphsthatmaximizeorminimizesomespectralparameter.
Of course, computing these eigenvalues and eigenvectors is a necessary step in
any such application. Since eigenvalues are the roots of a polynomial, in general
we cannot expect to find simple expressions for these roots. However, there are
numericalalgorithmsthatallowustoapproximatethemwithanydesiredprecision
inpolynomialtime.
In2011alineartimealgorithmwasdiscoveredfordecidinghowmanyeigenval-
uesoftheadjacencymatrixofatreelieinanyinterval.Inadditiontobeingfastand
easytoimplement,thisalgorithmissurprisinglysimpleand,atfirstglance,doesnot
seemtobeactingonmatrices.Itsunderlyingideawassoonextended todealwith
othermatricesandgraphclasses,givingrisetothesystematicstudyofwhatwenow
calleigenvaluelocationalgorithms.
Morethanbeingapracticalwayofapproximatingeigenvalues,thesealgorithms
haveshowntobeinstrumentaltosettletheoreticalquestionsinvolvingthedistribu-
tionofeigenvalues ingraphsinagiven class.Inparticular,aneigenvalue location
algorithmwasthemaintoolfortherecentsolutionofadifficultconjectureinvolving
thedistributionofLaplacianeigenvaluesintrees.
In this book, we survey the evolution of eigenvalue location algorithms in an
organizedandunifiedway,startingwithalgorithmsfortreesandotherwell-known
graph classes, such as cographs, and showing how they motivated more recent
vii
viii Preface
algorithms that may be applied to arbitrary graphs, but whose efficiency depends
ontheexistenceofagraphdecompositionoflowcomplexity.Whiletheyarevastly
deeperthanthesimpletreealgorithm,wewishtoconvincethereadersthattheyare
similarinspirit.
This book is intended for graduate students and researchers in spectral graph
theory. We have strived to make it as self-contained as possible. This has led
to a book that combines a compact introduction to spectral graph theory with a
discussion of structural decompositions such as tree decompositions and clique
decompositions,whichisrarelyexploredbybooksinthisarea.Wehopethatitmay
beusedasaconciseintroductiontoeigenvaluelocationalgorithmsforresearchers
whowishtoknowmoreabouteigenvaluesassociatedwithgraphs.
PortoAlegre,Brazil CarlosHoppen
Clemson,SC,USA DavidP.Jacobs
PortoAlegre,Brazil VilmarTrevisan
April2022
Acknowledgments
The authors are grateful to the Brazilian funding agencies CAPES (Coordenação
de Aperfeiçoamento de Pessoal de Nível Superior), CNPq (Conselho Nacional de
DesenvolvimentoCientíficoeTecnológico),andFAPERGS(FundaçãodeAmparo
àPesquisadoEstadodoRioGrandedoSul).Formanyyears,theseagencieshave
fostered the collaboration between Brazilian and international scientists from all
over the world. In particular, they funded several research visits leading to results
describedinthisbook.
Our thanks to SBMAC (Sociedade Brasileira de Matemática Aplicada e Com-
putacional)andtoSpringerNatureforsponsoringthisbook.Aspecialthankyouto
CelinadeFigueiredoandPaulo Silva,theeditorsofSBMACSpringer Briefs,and
toRobinsondosSantos,fromtheSãoPauloofficeofSpringerNature.
We are also thankful to three anonymous reviewers, whose careful reading and
thoughtfulconsiderationshaveledtoabetterbook.
ManyofthefiguresinthisbookweremadeusingTikZiT,aprogramfordrawing
graphs,makingtheconstructionofsuchfiguressimpleandstraightforward.
ix
Contents
1 Introduction .................................................................. 1
References .................................................................... 4
2 Preliminaries ................................................................. 5
2.1 GraphTheoryReview .................................................. 5
2.2 LinearAlgebraReview ................................................. 8
2.3 EigenvaluesandEigenvectors ......................................... 10
2.4 ElementaryMatricesandOperations .................................. 12
2.5 SpectralGraphTheory ................................................. 14
2.6 Sylvester’sLawofInertia .............................................. 17
2.7 AnalysisofAlgorithms ................................................ 19
2.8 RootedTrees ............................................................ 20
References .................................................................... 22
3 LocatingEigenvaluesinTrees.............................................. 25
3.1 AdjacencyMatrix ...................................................... 25
3.2 SymmetricMatriceswithUnderlyingTree ........................... 30
3.3 LaplacianMatrixandApplications .................................... 32
References .................................................................... 36
4 GraphClassesandGraphDecompositions ............................... 39
4.1 HereditaryGraphClasses .............................................. 39
4.2 Cographs ................................................................ 40
4.3 TreeDecomposition .................................................... 48
4.4 NiceTreeDecomposition .............................................. 51
4.5 CliqueDecomposition ................................................. 56
4.6 SlickCliqueDecomposition ........................................... 59
References .................................................................... 61
5 LocatingEigenvaluesinCographs......................................... 65
5.1 DiagonalizingaRowandColumn ..................................... 65
5.2 DiagonalizingA+xI .................................................. 69
xi
