Laplacian eigenvectors of graphs: Perron-Frobenius and Faber-Krahn type theorems

120 Pages·2007·0.901 MB·English

by  Türker Biyikoğu, Josef Leydold, Peter F. Stadler (auth.)

Lecture Notes in Mathematics (cid:49)(cid:57)(cid:49)(cid:53)
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
Türker Bıyıkog˘lu · Josef Leydold
Peter F. Stadler
Laplacian Eigenvectors
of Graphs
Perron-Frobenius and Faber-Krahn
Type Theorems
(cid:65)(cid:66)(cid:67)
Authors
TürkerBıyıkog˘lu JosefLeydold
DepartmentofMathematics DepartmentofStatisticsandMathematics
FacultyofArtsandSciences ViennaUniversityofEconomics
Is¸ıkUniversity andBusinessAdministration
S¸ile34980,Istanbul Augasse2-6
Turkey 1090Wien
e-mail:[email protected] Austria
URL:http://math.isikun.edu.tr/turker e-mail:[email protected]
URL:http://statmath.wu-wien.ac.at/˜leydold/
PeterF.Stadler
BioinformaticsGroup
DepartmentofComputerScience
UniversityofLeipzig
Härtelstrasse16-18
04107Leipzig
Germany
e-mail:[email protected]
URL:http://www.bioinf.uni-leipzig.de
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Preface
Eigenvectors of graph Laplacians are a rather esoteric topic for a book. In
fact, we are not aware of even a single review or survey article dedicated to
this topic. We have, however, two excuses: (1) There are fascinating subtle
differences between the properties of solutions of Schro¨dinger equations on
manifolds on the one hand, and their discrete analogs on graphs. (2) “Geo-
metric” properties of (cost) functions defined on the vertex sets of graphs
are of practical interest for heuristic optimization algorithms. Lov Grover’s
observation that the cost functions of quite a few of the well-studied combi-
natorialoptimizationproblemsareeigenvectorofassociatedgraphLaplacians
prompted us to investigate such eigenvectors more systematically.
The bookinessencecoverstwotopics:Nodaldomaintheoremswhichgive
bounds on the number of connected subgraphs on which an eigenvector does
notchangesign,andFaber-Krahn-typeinequalitieswhichareconcernedwith
the shape of domains (i.e., graphs in our setting) with fixed volume that
minimize the first Dirichlet eigenvalue. The connecting theme between these
two topics is focus on local and global properties of the eigenvectors (rather
than eigenvalues) and convenience of the Rayleigh quotient in the proofs.
The mindful readerwillfind thatmore oftenthan nota simple star graph
alreadyprovidesacounterexamplefor“obvious”conjectures.Infact,weused
the Petersen graph just because it seems against tradition to write about
graph theory without using the Petersen graph as a counterexample at least
once. The simplicity of the counterexamples highlights how little we know
aboutthe universe of graphLaplacianeigenvectors(and fitness landscapes in
general), and how misguided an intuition trained on well-behaved manifolds
canbeinthisrealm:evensmallmovesfrequentlycausesabrokennosecaused
by some unexpected wall.
The history of this monograph goes back more than a decade and has its
rootsintheinterdisciplinaryresearchenvironmentattheDepartmentofThe-
oreticalChemistry atthe UniversityofVienna,Austria.Acollaborationwith
Brian Davies during his stay at the Erwin Schro¨dinger Institute in Vienna
in 1995 stimulated our interest in Laplacian eigenvectors and eventually
VI Preface
resultedin a researchgrantfrom the AustrianFonds zur F¨orderung der Wis-
senschaftlichenForschung (projectno.14094-MAT)toinvestigatethistopicin
a more systematic way. Over the years, many colleagues contributed through
helpful discussions, among them Wim Hordijk, Ju¨rgen Jost, Bojan Mohar,
Tomaˇz Pisanski, Dan Rockmore, and Gerhard Wo¨ginger. We also thank the
Max Planck Institute for Mathematics in the Sciences in Leipzig for their
hospitality and for providing a fruitful scientific working for one of us (TB).
Leipzig, Tu¨rker Bıyıko˘glu
Wien, Josef Leydold
May 2006 Peter F. Stadler
Contents
1 Introduction............................................... 1
1.1 Matrix Representations of a Graph ........................ 2
1.2 Finite Differences........................................ 4
1.3 Landscapes on Graphs ................................... 4
1.4 Related Matrices ........................................ 7
1.5 Graphs with a Boundary: The Discrete Dirichlet Problem .... 8
1.6 Generalized Graph Laplacians............................. 10
1.7 Colin de Verdi`ere Matrices................................ 11
1.8 Practical Applications of Laplacians Eigenvectors............ 12
2 Graph Laplacians.......................................... 15
2.1 Basic Properties of Graph Laplacians ...................... 15
2.2 Weighted Graphs........................................ 17
2.3 The Rayleigh Quotient ................................... 18
2.4 Calculus on Graphs...................................... 19
2.5 Basic Properties of Eigenfunctions......................... 20
2.6 Graph Automorphisms and Eigenfunctions ................. 22
2.7 Quasi-Abelian Cayley Graphs ............................. 23
2.8 The Perron-Frobenius Theorem ........................... 26
3 Eigenfunctions and Nodal Domains ........................ 29
3.1 Courant’s Nodal Domain Theorem......................... 29
3.2 Proof of the Nodal Domain Theorem....................... 33
3.3 Algebraic Connectivity, Fiedler Vectors and Perron Branches.. 35
3.4 Some Results for Multiple Eigenvalues ..................... 39
3.5 The Courant-Herrmann Conjecture ........................ 41
3.6 Improvements for Special Cases ........................... 42
3.7 Faria Vectors and Minimum Number of Nodal Domains ...... 46
3.8 Sign Pattern and Nodal Domains.......................... 47
VIII Contents
4 Nodal Domain Theorems for Special Graph Classes........ 49
4.1 Trees .................................................. 49
4.2 Cographs and Threshold Graphs .......................... 54
4.3 Product Graphs and the Boolean Hypercube................ 58
5 Computational Experiments ............................... 67
5.1 Nodal Domains and Hyperplane Arrangements .............. 67
5.2 A Hillclimbing Algorithm................................. 69
5.3 Numerical Experiments for the Boolean Hypercube .......... 70
5.4 Local Optima ........................................... 73
6 Faber-Krahn Type Inequalities ............................ 77
6.1 Basic Properties of Dirichlet Operators..................... 78
6.2 The Faber-Krahn Property ............................... 79
6.3 Unweighted Trees ....................................... 81
6.4 Semiregular Trees ....................................... 84
6.5 Rearrangements and Dirichlet Operators ................... 86
6.6 Perturbations and Branches............................... 90
A Basic Notations............................................ 93
B Eigenfunctions Used in Figures ............................ 97
C List of Symbols ............................................ 99
References.....................................................101
Index..........................................................113
1
Introduction
The foundations of spectral graph theory were laid in the fifties and sixties
of the 20th century. The eigenvalues of graphs, most often defined as the
eigenvalues of the adjacency matrix, have since then received much attention
as a means of characterizing classes of graphs and for obtaining bounds on
properties such as the diameter, girth, chromatic number, connectivity [14,
17, 45, 46, 83, 85]. The interest has since then shifted somewhat from the
adjacency spectrum to the spectrum of the closely related graph Laplacian
[14, 35, 41, 85,88, 137, 139]. In particular,Laplaciangraphspectra arebeing
investigated as a means of characterizing large “small world networks” and
random graphs, see e.g. [33, 34, 119] for a few examples. For the most part,
the theory is still concerned with the eigenvalues.
Theeigenvectors ofgraphs,ontheotherhand,havereceivedonlysporadic
attention on their own, e.g. [134]. Even the book Eigenspaces of Graphs by
Cvetkovi´cet al. [47] contains only a few pages on the geometric properties of
the eigenvectors which are mostly used as a convenient proof technique.
In this book we will focus on mostly geometric properties of the eigenvec-
torsthemselves.Themotivationforthistopicistwofold.Asweshallseeinthis
firstintroductorychapter,theseobjectsariseinverydiverseapplications,from
mathematicalbiologytocombinatorialoptimization.TheLaplacianeigenvec-
tors are used as tools in heuristics to solve combinatorial problems on given
graphs, usually without a thorough understanding why they work so well.
From a more formal point of view, Laplacian eigenvectors are the natural
discretization of eigenfunctions of Laplace-Beltrami operators on manifolds.
Surprisingly, some of their properties in the discrete case are reminiscent of
corresponding results in the continuous setting, but often there are subtle
differences which we found interesting enough to explore in some detail.
We should, at this point, warn the reader: this book collects a number
of interesting facets of our topic, enough as we hope to stimulate further
research,butitcannotprovideacoherenttheoreticalframeworkorapowerful
machinery to tackle the properties of Laplacian eigenfunctions in generality.
2 1 Introduction
1.1 Matrix Representations of a Graph
Therearetwoobviouswaysofspecifyingasimple1 graphG(V,E)withvertex
setV ={1,...,n}andedgesetE bymeansofamatrix:theadjacencymatrix
and the incidence matrix. The adjacency matrix A has entries A = 1 if
xy
xy ∈ E and A = 0 otherwise: In order to specify the incidence matrix ∇
xy
we need an arbitrary but fixed orientation (direction) for each edge e = xy.
Then ∇ is a (|E|×|V|) matrix and has entries ∇ = −1 if x is the initial
ex
vertex of edge e, ∇ = 1 if x is the terminal vertex of edge e, and ∇ = 0
ex ex
otherwise, i.e., if x is not in e.
Let us now consider a real-valued function f over V, f: V → R. This is
simply a vector indexed by the vertices of G. In this book we prefer to use a
“functional”notationthatemphasizesthesimilaritiesbetweenthesituationof
graphsandmanifolds.Obviouslythesetofsuchfunctionsformsavectorspace
whichisisomorphictoRn (andthuswewill—byabuseofnotation—simply
denote this vector space by Rn). Similarly there exists a set of real-valued
functions over E. The mapf (cid:4)→∇f is known as the co-boundary mapping of
thegraphG.Itsvalue(∇f)(e)atagivenedgeeisthedifferenceofthevalues
of f at the two end-points of the edge e (considering orientation). Therefore
theincidencematrix∇isakindofdifferenceor“discretedifferential”operator
on G.
Let us now consider an Eulerian graph G. Recall that G is Eulerian if
and only if G is connected and all vertices have even degree. Let C be an
(arbitrary) Eulerian cycle in G (i.e., a closed walk that traverses each edge
exactly once) and fix an orientation of G such that C is properly oriented in
thesensethatalledgepoint“forward”alongC.ThecycleC maypassthrough
eachvertexxmultipletimes;theincomingedgeofthei-thpassise(cid:1) =(y(cid:1),x),
i i
the outgoing edge is e(cid:1)(cid:1) =(x,y(cid:1)(cid:1)). We can now define “2nd derivatives” along
i i
C:
(∂2 f)(x):=(∇f)(e(cid:1)(cid:1))−(∇f)(e(cid:1))
C;i
=[f(y(cid:1)(cid:1))−f(x)]−[f(x)−f(y(cid:1))]
i i
=f(y(cid:1))+f(y(cid:1)(cid:1))−2f(x).
i i
Notethat(∂2 f)(x)isindependentoftheorientationonG.Interpretingeach
C;i
pass of C through x as a different “dimension” it seems natural to consider
the sum over these “2nd derivatives” as a “Laplace-Beltramioperator”
(cid:1)
(∆f)(x)= (∂2 f)(x)
C;i
passesiofC throughx
d((cid:1)x)/2 (cid:1)
= [f(y(cid:1))+f(y(cid:1)(cid:1))−2f(x)]= [f(y)−f(x)]
i i
i=1 y∼x
1 For basic definitions and results thereader is referred to AppendixA.
1.1 Matrix Representations of a Graph 3
whichisindependentofthechoiceoftheEuleriancycleC andtheorientation
on G. Naturally, one generalizes this definition of ∆ to arbitrary graphs.
Inthegraphtheoryliterature,however,itiscustomarytodefinetheLapla-
cian operator (map) L: R|V| →R|V| with the opposite sign:
(cid:1)
(Lf)(x)=(−∆f)(x)= [f(x)−f(y)]. (1.1)
y∼x
From an algebraic point of view it appears more natural to define
L=∇T∇ (1.2)
which is known as the Laplacian (matrix) of G. We have
⎧
(cid:1) ⎪⎨−1 if xy ∈E,
L = ∇ ∇ = d(x) if x=y, (1.3)
xy ex ey ⎪⎩
e∈E 0 otherwise,
where d(x)=|{e∈E|x∈e}| is the degree of the vertex x. It is important to
note that L in (1.3) is independent of the orientation of the edges. Clearly,
xy
we have the identity (Lf)(x)=(Lf)(x).
Defining the diagonal matrix D with entries D = d(x), called the de-
xx
gree matrix, we obtain a simple connection between the Laplacian and the
adjacency matrix of a graph,
L=D−A. (1.4)
The Laplacian L therefore uniquely determines its graph through its off-
diagonal entries.
Thecloserelationbetween∇andLontheonehand,andtheirdifferential
operator counterparts on the other hand, is exemplified by the following dis-
creteversionofGreen’sformula,whichiseasilyverifiedbydirectcomputation
[159]:
Proposition 1.1.Let f: V → R and g : V → R be two arbitrary functions.
Then
(cid:1) (cid:1) (cid:1)
f(x)(Lg)(x)= g(x)(Lf)(x)= (∇f)(e)(∇g)(e).
x∈V x∈V e∈E
Using angularbrackets(cid:5)·,·(cid:6) to denote the usualscalarproductoftwo vectors
in Rn and the symbol −∆ for L we can formulate Green’s formula in a more
familiar form as
−(cid:5)f,∆g(cid:6)=−(cid:5)∆f,g(cid:6)=(cid:5)∇f,∇g(cid:6).
The Laplacian L can thus be viewed as a proper discretization of the usual
Laplace-Beltramidifferential operator.