29
Selected Title s i n Thi s Serie s
92 Fa n R. K . Chung , Spectra l grap h theory , 199 7
91 J . P . Ma y et al. , Equivarian t homotop y an d cohomolog y theory , dedicate d t o th e memor y
of Rober t J . Piacenza , 199 6
90 Joh n Roe , Inde x theory , coars e geometry , an d topolog y o f manifolds , 199 6
89 Cliffor d Henr y Taubes , Metrics , connection s an d gluin g theorems , 199 6
88 Crai g Huneke , Tigh t closur e an d it s applications , 199 6
87 Joh n Eri k Fornaess , Dynamic s i n severa l comple x variables , 199 6
86 Sori n Popa , Classificatio n o f subfactor s an d thei r endomorphisms , 199 5
85 Michi o Jimbo an d Tetsuj i Miwa , Algebrai c analysi s o f solvabl e lattic e modeis , 199 4
84 Hug h L . Montgomery, Te n lecture s o n th e interfac e betwee n analyti c numbe r theor y an d
harmonic analysis , 1994
83 Carlo s E . Kenig , Harmoni c analysi s technique s fo r secon d orde r ellipti c boundar y valu e
Problems, 199 4
82 Susa n Montgomery , Hop f algebra s an d thei r action s o n rings , 199 3
81 Steve n G . Krantz, Geometri e analysi s an d funetio n spaces , 199 3
80 Vaugha n F . R. Jones , Subfactor s an d knots , 199 1
79 Michae l Frazier , Björ n Jawerth, an d Guid o Weiss , Littlewood-Pale y theor y an d th e stud y
of funetio n spaces , 199 1
78 Edwar d Formanek , Th e polynomia l identitie s an d variant s o f n x n matrices , 199 1
77 Michae l Christ , Lecture s o n singula r integra l Operators , 199 0
76 Klau s Schmidt , Algebrai c idea s i n ergodi c theory , 199 0
75 F . Thoma s Farrel l and L. Edwin Jones, Classica l aspherica l manifolds , 199 0
74 Lawrenc e C . Evans , Wea k convergenc e method s fo r nonlinea r partia l differentia l
equations, 199 0
73 Walte r A . Strauss , Nonlinea r wav e equations , 198 9
72 Pete r Orlik , Introductio n t o arrangements , 198 9
71 Harr y Dym , J contractiv e matri x funetions , reproducin g kerne l Huber t space s an d
interpolation, 198 9
70 Richar d F . Gundy , Som e topic s i n probabilit y an d analysis , 198 9
69 Fran k D . Grosshans , Gian-Carl o Rota , an d Joe l A . Stein , Invarian t theor y an d
superalgebras, 198 7
68 J . Willia m Heiton , Josep h A . Ball , Charle s R . Johnson , an d Joh n N . Palmer , Operator theory , analyti c funetions , matrices , an d electrica l engineering , 198 7
67 Haral d Upmeier , Jorda n algebra s i n analysis , Operato r theory , an d quantu m mechanics , 1987
66 G . Andrews , g-Series : Thei r developmen t an d applicatio n i n analysis , numbe r theory ,
combinatorics, physic s an d Compute r algebra , 198 6
65 Pau l H . Rabinowitz , Minima x method s i n critica l poin t theor y wit h application s t o
differential equations , 198 6
64 Donal d S . Passman , Grou p rings , crosse d produet s an d Galoi s theory , 198 6
63 Waite r Rudin , Ne w construetion s o f funetion s holornorphi c i n th e uni t bal l o f C n , 198 6
62 Bel a Bollobas , Extrema l grap h theor y wit h emphasi s o n probabilisti c methods , 198 6
61 Mogen s Flensted-Jensen , Analysi s o n non-Riemannia n Symmetri e spaces , 198 6
60 Gille s Pisier , Factorizatio n o f linea r Operator s an d geometr y o f Banac h spaces , 198 6
59 Roge r How e an d Allen Moy , Harish-Chandr a homomorphism s fo r p-adi c groups , 198 5
58 H . Blaine Lawson , Jr. , Th e theor y o f gaug e fields i n fou r dimensions , 198 5
57 Jerr y L . Kazdan, Prescribin g th e curvatur e o f a Riemannia n manifold , 198 5 (Continued in the back of this publication)
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Spectral Graph Theoiy
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Conference Boar d o f the Mathematica l Science s
CBMS Regional Conference Serie s in Mathematics
Number 9 2
Spectral Graph Theoiy
Fan R . K . Chun g
Published fo r th e Conference Boar d o f the Mathematica l Science s
by th e ^ = ^ v America n Mathematica l Societ y ^ük5 %
Providence, Rhod e Islan d ^GRSR- * with suppor t fro m th e v^SaF** "
*^sr^ Nationa l Scienc e Foundatio n 0 **o^0
http://dx.doi.org/10.1090/cbms/092
CBMS Conferenc e o n Recen t Advance s in Spectra l Grap h Theor y
held a t Californi a Stat e Universit y a t Fresno , June 6-10 , 199 4
Research partiall y supporte d b y th e National Scienc e Foundatio n
2000 Mathematics Subject Classification. Primar y 05-XX .
Library o f Congres s Cataloging-in-Publicat io n Dat a
Chung, Fa n R . K . Spectral grap h theor y / Fa n R . K . Chung .
p. cm . — (Regiona l Conferenc e serie s i n mathematics , ISS N 0160-764 2 ; no. 92 ) "CBMS Conferenc e o n Recen t Advance s i n Spectra l Grap h Theor y hel d a t Californi a Stat e
University a t Fresno , Jun e 6-10 , 1994"—T.p . verso . Includes bibliographica l references . ISBN 0-8218-0315- 8 (sof t : alk . paper ) 1. Grap h theory—Congresses . 2 . Eigenvalues—Congresses . I . CBM S Conferenc e o n Recen t
Advances i n Spectra l Grap h Theor y (199 4 : Californi a Stat e Universit y a t Fresno ) I L Title . III. Series .
QA1.R33 no . 9 2 [QA166] 511'.5—dc21 96-4511 2
CIP
Copying an d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them , ar e permitte d t o mak e fai r us e o f th e material , suc h a s t o cop y a chapte r fo r us e in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customar y acknowledgmen t o f th e sourc e i s given .
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Contents
Preface x i
Chapter 1 . Eigenvalue s an d th e Laplacia n o f a graph 1
1.1. Introductio n 1
1.2. Th e Laplacia n an d eigenvalue s 2
1.3. Basi c fact s abou t th e spectru m o f a grap h 6
1.4. Eigenvalue s o f weighted graph s 1 1
1.5. Eigenvalue s an d rando m walk s 1 4
Chapter 2 . Isoperimetri c problem s 2 3
2.1. Histor y 2 3
2.2. Th e Cheege r constan t o f a graph 2 4
2.3. Th e edg e expansio n o f a graph 2 5
2.4. Th e verte x expansio n o f a graph 2 9
2.5. A characterization o f the Cheege r constan t 3 2
2.6. Isoperimetri c inequalitie s fo r cartesia n product s 3 6
Chapter 3 . Diameter s an d eigenvalue s 4 3
3.1. Th e diarnete r o f a graph 4 3
3.2. Eigenvalue s an d distance s betwee n tw o subset s 4 5
3.3. Eigenvalue s an d distance s amon g man y subset s 4 9
3.4. Eigenvalu e uppe r bound s fo r manifold s 5 0
Chapter 4 . Paths , flows, and routin g 5 9
vni CONTENT S
4.1. Path s an d set s o f path s 5 9
4.2. Flow s an d Cheege r constant s 6 0
4.3. Eigenvalue s an d route s wit h smal l congestio n 6 2
4.4. Routin g i n graph s 6 4
4.5. Compariso n theorem s 6 8
Chapter 5 . Eigenvalue s an d quasi-randomnes s 7 3
5.1. Quasi-randomnes s 7 3
5.2. Th e discrepanc y propert y 7 5
5.3. Th e deviatio n o f a graph 8 1
5.4. Quasi-rando m graph s 8 5
Chapter 6 . Expander s an d explici t construction s 9 1
6.1. Probabilisti c method s versu s explici t construction s 9 1
6.2. Th e expander s 9 2
6.3. Example s o f explici t construction s 9 7
6.4. Application s o f expanders i n communicatio n network s 10 2
6.5. Construction s o f graph s wit h smal l diamete r an d girt h 10 5
6.6. Weighte d Laplacian s an d th e Lovas z d functio n 10 7
Chapter 7 . Eigenvalue s o f symmetrica l graph s 11 3
7.1. Symmetrica l graph s 11 3
7.2. Cheege r constant s o f symmetrica l graph s 11 4
7.3. Eigenvalue s o f symmetrica l graph s 11 6
7.4. Distanc e transitiv e graph s 11 8
7.5. Eigenvalue s an d grou p representatio n theor y 12 1
7.6. Th e vibrationa l spectru m o f a graph 12 3
Chapter 8 . Eigenvalue s o f subgraph s wit h boundar y condition s 12 7
8.1. Neuman n eigenvalue s an d Dirichle t eigenvalue s 12 7
CONTENTS i x
8.2. Th e Neuman n eigenvalue s o f a subgraph 12 8
8.3. Neuman n eigenvalue s an d rando m walk s 13 0
8.4. Dirichle t eigenvalue s 13 2
8.5. A matrix-tre e theore m an d Dirichle t eigenvalue s 13 3
8.6. Determinant s an d invarian t field theor y 13 5
Chapter 9 . Harnac k inequalitie s 13 9
9.1. Eigenfunction s 13 9
9.2. Conve x subgraph s o f homogeneou s graph s 14 0
9.3. A Harnac k inequalit y fo r homogeneou s graph s 14 2
9.4. Harnac k inequalitie s fo r Dirichle t eigenvalue s 14 4
9.5. Harnac k inequalitie s fo r Neuman n eigenvalue s 14 6
9.6. Eigenvalue s an d diameter s 14 8
Chapter 10 . Hea t kernei s 14 9
10.1. Th e hea t kerne l o f a grap h an d it s induce d subgraph s 14 9
10.2. Basi c fact s o n hea t kernei s 15 0
10.3. A n eigenvalu e inequalit y 15 2
10.4. Hea t kerne l lowe r bound s 15 4
10.5. Matrice s wit h give n ro w an d colum n sum s 16 0
10.6. Rando m walk s an d th e hea t kerne l 16 5
Chapter 11 . Sobole v inequalitie s 16 7
11.1. Th e isoperimetri c dimensio n o f a graph 16 7
11.2. A n isoperimetri c inequalit y 16 9
11.3. Sobole v inequalitie s 17 2
11.4. Eigenvalu e bound s 17 4
11.5. Generalization s t o weighte d graph s an d subgraph s 17 9
Chapter 12 . Advance d technique s fo r rando m walk s o n graph s 181
x CONTENT S
12.1. Severa l approache s fo r boundin g convergenc e 18 1
12.2. Logarithmi c Sobole v inequalitie s 18 4
12.3. A comparison theore m fo r th e log-Sobole v constan t 18 9
12.4. Logarithmi c Harnac k inequalitie s 19 1
12.5. Th e isoperimetri c dimensio n an d th e Sobole v inequalit y 19 6
Bibliography 20 1
Index 21 0
Preface
This monograp h i s an intertwine d tal e o f eigenvalues an d thei r us e i n unlock -ing a thousand secret s abou t graphs . Th e storie s wil l b e tol d — ho w the spectru m reveals fundamenta l propertie s o f a graph , ho w spectra l grap h theor y link s th e discrete univers e t o th e continuou s on e throug h geometric , analyti c an d algebrai c techniques, an d how , throug h eigenvalues , theor y an d application s i n Communica -tions an d Compute r scienc e com e togethe r i n symbioti c harmony . Sinc e spectra l graph theor y ha s bee n evolvin g very rapidly , th e abov e goal s ca n onl y b e partiall y fulfilled here . Fo r example , mor e advance d materia l o n th e hea t kerne l wil l b e treated i n a later publication ,
This boo k i s base d o n te n lecture s give n a t th e CBM S Worksho p o n spectra l graph theor y i n Jun e 199 4 a t Fresn o Stat e University . Additiona l chapter s wer e included o n subgraph s wit h boundar y condition s an d o n advance d technique s fo r random walks . I wish t o than k S.T . Ya u fo r introducin g m e t o powerfu l idea s fro m spectral geometry . Th e las t fe w chapter s ar e mainl y base d o n ou r collaborativ e work o n geometr y o f graphs . Specia l thank s ar e du e t o Ro n Graham , withou t whose encouragement thi s book would never have existed. I n the course of writing, I have greatly benefitte d fro m numerou s suggestion s an d valuabl e comment s o f Noga Alon, And y Woldar , Feli x Lazebnik , Davi d Gillman , Elizabet h Wilmer , Prasa d Tetali an d Her b Wilf . I woul d als o lik e t o acknowledg e th e suppor t o f Nationa l Science Foundatio n throug h Gran t No . DM S 95-04834 .
Like al l authors , I woul d hop e tha t thes e lectur e note s ar e completel y error -free. However , t o b e realistic , I pla n t o maintai n a n errat a lis t o n m y hom e pag e http://www.math.upenn.edu/~chung. Naturally , I wil l b e gratefu l fo r an y contri -bution t o thi s list .
Fan Chun g
September, 199 6
XI
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Index
(7r;p)-norm, 18 5 Z/2-norm, 1 5 X-squared distance , 1 9 /c-access graph , 10 2 n-cube, 20 , 3 7 p-norm, 4 1
abelian group , 14 0 adjacency matrix , 3 adjacent, 3 all distance s algorithm , 4 4 all shortes t path s algorithm , 4 4 almost regulär , 8 2 aperiodicity, 1 4 automorphism, 2 0 automorphism group , 11 4
bipartite, 6 , 7 bipartite expande r graph , 9 6 bipartite graph , 9 6 bond, 12 2 boundary conditions , 17 9 boundary expansio n property , 14 1 boundary Operator , 3 Buckyball, 12 2 Buckyball graph , 11 3
cartesian product , 37 , 6 6 Cayley graphs , 11 4 chain, 3 Chebyshev polynomial , 44 , 4 6 Cheeger constant , 24 , 25 , 60 , 93 , 11 5
characterization, 3 2 modified, 35 , 3 6 weighted graphs , 3 6
Cheeger inequality , 2 6 chromatic number , 10 8
clique, 81 , 91 , 11 0 clique number , 11 0 co-NP-complete, 9 4 coboundary Operator , 3 comparison theorems , 6 8 complete graph , 6 concentrator, 9 3 conductance, 9 4 connected, 7 , 2 4 consistent, 3 6 contingency tabl e problem , 16 0 contraction, 1 3 convex subgraph , 141 , 15 4 coset graph , 9 9 cycle, 6
degree, 2 deviation, 8 2 diameter, 8 , 43 , 10 6 diameter algorithm , 4 4 diameter-eigenvalue inequalities , 4 4 Dirichlet boundar y condition , 127 , 151 , 19 4 Dirichlet eigenvalues , 127 , 132-134 , 136 , 15 4 Dirichlet sum , 4 discrepancy, 75 , 78 , 8 6 distance, 8 , 4 5 distance transitive , 113 , 118 , 12 0
edge boundary , 24 , 12 7 edge expansion , 2 5 edge generatin g set , 114 , 15 5 edge transitive , 11 6 edge-cut, 2 3 edge-transitive, 11 4 eigenfunction, 4
harmonic, 4 eigenvalue, 4
210
INDEX 211
entropy function , 10 5 ergodic, 1 4 expander coefficient , 9 3 expander graph , 91-9 3 expansion factor , 9 3 explicit construction , 91 , 97
flow, 5 9 Fourier coefficients , 4 5
girth, 10 7 graph, 2
weighted undirected , 1 2 group representatio n theory , 12 1
Hamiltonian path , 5 9 Hamming balls , 3 7 Harnack inequality , 139 , 14 2 heat equation , 14 9 heat kernel , 14 9 heat kerne l eigenvalu e inequality , 15 2 homogenous graphs , 12 3 Hooke's law , 12 4 hypercube, 37 , 19 9
independent set , 8 1 induced subgraph , 12 7 inner product , 4 intersection graph , 12 1 invariant, 14 0 irreducibility, 1 4 isoperimetric dimension , 167 , 18 3 isoperimetric inequalities , 16 9 isoperimetric number , 35 , 9 4 isotropy group , 11 4
Laplace Operator , 50 , 53 , 15 7 Laplace-Beitrarni Operator , 4 Laplacian, 2
determinant o f , 13 5 lattice graphs , 15 5 lazy walk , 16 , 18 , 18 1 local fc-frame, 19 9 log-Sobolev constant , 181 , 18 7
with Dirichle t boundar y condition , 18 3 with Neuman n boundar y condition , 18 3
logarithmic Harnac k inequalities , 18 2 logarithmic Sobole v inequalities , 22 , 18 4 loop, 2
major acces s network , 10 2 Margulis graph , 9 9 matching, 104 , 10 6 matrix multiplication , 4 4 matrix-tree theorem , 13 3 measure, 4 , 2 4 moderate growt h rate , 16 9 Moore bound , 10 6
neighborhood, 74 , 93 , 94 , 19 9
Neumann boundar y condition , 127 , 150 , 19 4 Neumann eigenvalues , 127 , 130 , 15 4 non-blocking network , 10 2 normal, 9 1 NP, 11 2 NP-complete, 5 9
odd-even transpositio n sort , 6 5 orthonormal labeling , 11 1
pairing, 6 7 Paley graph , 81 , 97 Paley su m graph , 9 8 partition function , 13 5 path, 6 perfect graph , 11 0 Petersen graph , 12 1 Plancherei formula , 2 0 Poincare inequalities , 6 8 polynomial growt h rat e , 16 9 probabilistic method , 9 1 projection, 14 9
quasi-random, 7 3 quaternion group , 10 1
Raman spectrum , 12 5 Ramanujan graph , 44 , 101 , 10 7 Ramsey property , 8 0 random graph , 11 2 random walk , 14 , 6 2
cartesian produc t o f graphs , 3 7 ergodic, 1 4 irreducible, 1 4 modified, 1 6 Neumann, 13 0 reversible, 1 4
Rayleigh quotient , 4 regulär, 3 relative pointwis e distance , 17 , 165 , 18 1 Ricci flat, 19 9 Riemannian manifold , 4 rooted spannin g forest , 133 , 13 6 route coverin g number , 6 7 route set , 5 9 routing, 5 9
complete bipartit e graph , 6 5 complete graph , 6 5 hypercube, 65 , 6 7 tree, 6 5
routing assignment , 6 4 routing number , 6 4
Separator, 2 3 size, 2 3 Sobolev constant , 41 , 167 , 16 8 Sobolev inequalities , 16 7 spanning tree , 13 3 spectrum, 4 star, 6
212
stationary distribution , 1 4 strongly conve x subgraph , 140 , 19 4 support, 5 1 Symmetrie group , 6 4
total Variatio n distance , 1 8 trace, 1 0 transpose, 3 , 1 7 transposition, 6 4
Vertex boundary , 24 , 12 7 Vertex expansion , 2 4 Vertex transitive , 113 , 114 , 11 7 Vertex weight , 4 vertex-cut, 2 3 vertex-transitive, 11 4 vertex-transitive graph , 2 0 vibrational Laplacian , 12 3 vibrational spectrum , 12 3 volume, 5 Voronoi region , 15 8
walk, 1 4 weight funetion , 3 6 weighted cartesia n product , 18 4 weighted graph , 165 , 17 9
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Selected Title s i n Thi s Serie s (Continued from the front of this publication)
56 Har i Bercovici, Cipria n Foias., and Carl Pearcy, Dua l algebra s wit h application s t o invarian t subspaces an d dilatio n theory , 198 5
55 Willia m Arveson , Te n lecture s o n Operato r algebras , 198 4
54 Willia m Fulton , Introductio n t o intersectio n theor y i n algebrai c geometry , 198 4
53 Wilhel m Klingenberg , Close d geodesic s o n Riemannia n manifolds , 198 3
52 Tsit-Yue n Lam , Orderings , valuation s an d quadrati c forms , 198 3
51 Masamich i Takesaki , Structur e o f factor s an d automorphis m groups , 198 3
50 Jame s Eell s and Luc Lemaire , Selecte d topic s i n harmoni c maps , 198 3
49 Joh n M. Franks , Homolog y an d dynamica l Systems , 198 2
48 W . Stephe n Wilson , Brown-Peterso n homology : a n introductio n an d sampler , 198 2
47 Jac k K . Haie , Topic s i n dynami c bifurcatio n theory , 198 1
46 Edwar d G . Effros , Dimension s an d C*-algebras , 198 1
45 Ronal d L . Graham, Rudiment s o f Ramse y theory , 198 1
44 Philli p A . Griffiths , A n introductio n t o th e theor y o f specia l divisor s o n algebrai c curves ,
1980
43 Willia m Jaco, Lecture s o n three-manifol d topology , 198 0
42 Jea n Dieudonne , Specia l function s an d linea r representation s o f Li e groups , 198 0
41 D . J . Newman , Approximatio n wit h rationa l functions , 197 9
40 Jea n Mawhin , Topologica l degre e method s i n nonlinea r boundar y valu e problems , 197 9
39 Georg e Lusztig , Representation s o f finite Chevalle y groups , 197 8
38 Charle s Conley , Isolate d invarian t set s an d th e Mors e index , 197 8
37 Masayosh i Nagata , Polynomia l ring s an d affin e Spaces , 197 8
36 Car l M . Pearcy , Som e recen t development s i n Operato r theory , 197 8
35 R . Bowen , O n Axio m A difFeomorphisms , 197 8
34 L . Auslander, Lectur e note s o n nil-thet a functions , 197 7
33 G . Glauberman , Factorization s i n loca l subgroup s o f finite groups , 197 7
32 W . M . Schmidt , Smal l fractiona l part s o f polynomials , 197 7
31 R . R . Coifma n an d G. Weiss , Transferenc e method s i n analysis , 197 7
30 A . Pelczyriski , Banac h Space s o f analyti c function s an d absolutel y summin g Operators ,
1977
29 A . Weinstein , Lecture s o n symplecti c manifolds , 197 7
28 T . A . Chapman , Lecture s o n Huber t cub e manifolds , 197 6
27 H . Blain e Lawson , Jr., Th e quantitativ e theor y o f foliations , 197 7
26 I . Reiner , Clas s group s an d Picar d group s o f grou p ring s an d Orders , 197 6
25 K . W . Gruenberg , Relatio n module s o f finite groups , 197 6
24 M . Höchster , Topic s i n th e homologica l theor y o f module s ove r commutativ e rings , 197 5
23 M . E . Rudin , Lecture s o n se t theoreti c topology , 197 5
22 O . T. O'Meara , Lecture s o n linea r groups , 197 4
21 W . Stoll , Holomorphi c function s o f finite orde r i n severa i compie x variables , 197 4
20 H . Bass , Introductio n t o som e method s o f algebrai c K-theory , 197 4
19 B . Sz.-Nagy , Unitar y dilation s o f Huber t spac e Operator s an d relate d topics , 197 4
18 A . Friedman, Differentia l games , 197 4
17 L . Nirenberg, Lecture s o n linea r partia l differentia l equations , 197 3
16 J . L . Taylor, Measur e algebras , 197 3
15 R . G . Douglas , Banac h algebr a technique s i n th e theor y o f Toeplit z Operators , 197 3
(See th e AM S catalo g fo r earlie r titles )
