A Brief Introduction to Spectral Graph Theory

167 Pages·2018·1.09 MB·English

by  Bogdan Nica

Textbooks in Mathematics
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Bogdan Nica
Bogdan Nica
A Brief Introduction to
A Brief Introduction
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Spectral Graph Theory

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Spectral graph theory starts by associating matrices to graphs – notably, the adjacency t
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matrix and the Laplacian matrix. The general theme is then, firstly, to compute or o
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estimate the eigenvalues of such matrices, and secondly, to relate the eigenvalues to u
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structural properties of graphs. As it turns out, the spectral perspective is a powerful t Theory
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tool. Some of its loveliest applications concern facts that are, in principle, purely graph
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theoretic or combinatorial. t
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This text is an introduction to spectral graph theory, but it could also be seen as an p
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invitation to algebraic graph theory. The first half is devoted to graphs, finite fields, and c
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how they come together. This part provides an appealing motivation and context of the ra
second, spectral, half. The text is enriched by many exercises and their solutions. l
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The target audience are students from the upper undergraduate level onwards. We p
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assume only a familiarity with linear algebra and basic group theory. Graph theory,
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finite fields, and character theory for abelian groups receive a concise overview and h
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render the text essentially self-contained. o
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ISBN 978-3-03719-188-0
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Bogdan Nica
A Brief Introduction
to Spectral
Graph Theory
Author:
Bogdan Nica
Department of Mathematics and Statistics
McGill University
805 Sherbrooke St W
Montreal, QC H3A 0B9
Canada
E-mail: [email protected]
2010 Mathematics Subject Classification: Primary: 05-01, 05C50; secondary: 05C25, 11T24, 15A42
Key words: Adjacency eigenvalues of graphs, Laplacian eigenvalues of graphs, Cayley graphs,
algebraic graphs over finite fields, character sums
ISBN 978-3-03719-188-0
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Contents
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1 Graphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Bipartitegraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Invariants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Chromaticnumberandindependencenumber . . . . . . . . . . . . 11
2.2 Diameterandgirth . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Isoperimetricnumber . . . . . . . . . . . . . . . . . . . . . . . . . 15
3 Regulargraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Cayleygraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Cayleygraphs,continued . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Stronglyregulargraphs . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Designgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4 Finitefields. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.1 Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2 Projectivecombinatorics . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 Incidencegraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5 Squaresinfinitefields. . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.1 Thequadraticcharacter . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Quadraticreciprocity . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.3 Paleygraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4 Acomparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6 Characters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.1 Charactersoffiniteabeliangroups . . . . . . . . . . . . . . . . . . 51
6.2 Charactersumsoverfinitefields . . . . . . . . . . . . . . . . . . . 53
6.3 Morecharactersumsoverfinitefields . . . . . . . . . . . . . . . . 57
6.4 AnapplicationtoPaleygraphs . . . . . . . . . . . . . . . . . . . . 59
viii Contents
7 Eigenvaluesofgraphs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
7.1 AdjacencyandLaplacianeigenvalues . . . . . . . . . . . . . . . . 63
7.2 Firstproperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
7.3 Firstexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
8 Eigenvaluecomputations. . . . . . . . . . . . . . . . . . . . . . . . . . . 73
8.1 Cayleygraphsandbi-Cayleygraphsofabeliangroups . . . . . . . . 73
8.2 Stronglyregulargraphs . . . . . . . . . . . . . . . . . . . . . . . . 76
8.3 Twogems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
8.4 Designgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
9 Largesteigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
9.1 Extremaleigenvaluesofsymmetricmatrices . . . . . . . . . . . . . 85
9.2 Largestadjacencyeigenvalue . . . . . . . . . . . . . . . . . . . . . 86
9.3 Theaveragedegree . . . . . . . . . . . . . . . . . . . . . . . . . . 88
9.4 AspectralTurántheorem . . . . . . . . . . . . . . . . . . . . . . . 91
9.5 LargestLaplacianeigenvalueofbipartitegraphs . . . . . . . . . . . 93
9.6 Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
9.7 Largesteigenvaluesoftrees . . . . . . . . . . . . . . . . . . . . . . 97
10 Moreeigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
10.1 Eigenvaluesofsymmetricmatrices: Courant–Fischer . . . . . . . . 103
10.2 AboundfortheLaplacianeigenvalues . . . . . . . . . . . . . . . . 104
10.3 Eigenvaluesofsymmetricmatrices: CauchyandWeyl. . . . . . . . 107
10.4 Subgraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11 Spectralbounds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
11.1 Chromaticnumberandindependencenumber . . . . . . . . . . . . 113
11.2 Isoperimetricconstant. . . . . . . . . . . . . . . . . . . . . . . . . 116
11.3 Edgecounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
12 Farewell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
12.1 Graphswithout4-cycles . . . . . . . . . . . . . . . . . . . . . . . 127
12.2 TheErdős–Rényigraph . . . . . . . . . . . . . . . . . . . . . . . . 128
12.3 EigenvaluesoftheErdős–Rényigraph . . . . . . . . . . . . . . . . 130
Furtherreading. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Solutionstoexercises. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Introduction
Spectral graph theory starts by associating matrices to graphs — notably, the adja-
cencymatrixandtheLaplacianmatrix. Thegeneralthemeisthen,firstly,tocompute
or estimate the eigenvalues of such matrices, and secondly, to relate the eigenval-
ues to structural properties of graphs. As it turns out, the spectral perspective is a
powerfultool. Someofitsloveliestapplicationsconcernfactsthatare,inprinciple,
purelygraphtheoreticorcombinatorial. Togivejustoneexample,spectralideasare
a key ingredient in the proof of the so-called friendship theorem: if, in a group of
people,anytwopersonshaveexactlyonecommonfriend,thenthereisapersonwho
iseverybody’sfriend.
Thistextisanintroductiontospectralgraphtheory,butitcouldalsobeseenasan
invitationtoalgebraicgraphtheory. Ontheonehand, thereis, ofcourse, thelinear
algebrathatunderliesthespectralideasingraphtheory. Ontheotherhand,mostof
our examples are graphs of algebraic origin. The two recurring sources are Cayley
graphs of groups, and graphs built out of finite fields. In the study of such graphs,
somefurtheralgebraicingredients—forexample,characters—naturallycomeup.
The table of contents gives, as it should, a good glimpse of where this text is
going. Very broadly, the first half is devoted to graphs, finite fields, and how they
come together. This part is meant as an appealing and meaningful motivation. It
providesacontextthatframesandfuelsmuchofthesecond,spectral,half.
Many sections have one or two exercises. They are optional, in the sense that
virtuallynothinginthemainbodydependsonthem. Buttheexercisesareusuallyof
the non-trivial variety, and they should enhance the text in an interesting way. The
hopeisthatthereaderwillenjoythem. Atanyrate,solutionsareprovidedattheend
ofthetext.
We assume a basic familiarity with linear algebra, finite fields, and groups, but
notnecessarilywithgraphtheory. This,again,betraysouralgebraicperspective.
Acknowledgments. IwouldliketothankPéterCsikvári,forthoughtfulfeedback;
Sebastian Cioabă, for several discussions; Jerome Baum, for early help with some
pictures;AlainMatthes,forhisonlinegalleryofnamedgraphs.