Eigenvalues and
geometric representations
of graphs
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA 98052
Every planar graph can be drawnin the plane with straight edges
Fáry-Wagner
planar graph
Steinitz 1922
Every 3-connected planar graphis the skeleton of a convex 3-polytope.
3-connected planar graph
Rubber bands and planarity
Every 3-connected planar graph can be drawn with straight edges and convex faces.
Tutte (1963)
Rubber bands and planarity
outer face fixed toconvex polygon
edges replaced byrubber bands
2( )i jij E
u u
EEnergy:
Equilibrium:( )
1i j
j N ii
u ud
Demo
G 3-connected planar
rubber band embedding is planar
Tutte
(Easily) polynomial time
computable
Maxwell-Cremona
Lifts to Steinitz representation if
outer face is a triangle
If the outer face is a triangle, then the Tutte representationis a projection of a Steinitz representation.
1Fv
If the outer face is a not a triangle, then go to the polar.
F F’
p’
p
' '( )p p F Fu u v v ¿- = -
: 0polytope,dP P
* { : 1 }: polar polytoped TP y x y x P
G: arbitrary graph
A,B V, |A|=|B|=3
A: triangle
edges: rubber bands
2( )ij i jij E
c u u
EEnergy:
Equilibrium:( )
iji j
j N i i
cu u
d
( )
i ijj N i
d c
Rubber bands and connectivity
()
For almost all choices of edge strengths:B noncollinear
3 disjoint (A,B)-paths
Linial-L-Wigderson
cutset
For almost all choices of edge strengths:B affine indep
k disjoint (A,B)-paths
Linial-L-Wigderson
()
strengthen
edges strength s.t. B is independent
for a.a. edge strength, B is independent
no algebraic relationbetween edge strength
The maximum cut problem
maximize
Applications: optimization, statistical mechanics…
Easy with 50% error Erdős
Bad news: Max Cut is NP-hard
Approximations?
C
C
Easy with 50% error Erdős
Bad news: Max Cut is NP-hard
Approximations?
Easy with 50% error Erdős
Bad news: Max Cut is NP-hard
Approximations?
Easy with 50% error Erdős
Bad news: Max Cut is NP-hard
Approximations?
Easy with 50% error Erdős
Bad news: Max Cut is NP-hard
Approximations?
Easy with 50% error Erdős
Bad news: Max Cut is NP-hard
Approximations?
Easy with 50% error Erdős
Bad news: Max Cut is NP-hard
Approximations?
NP-hard with 6% error Hastad
Polynomial with 12% error
Goemans-Williamson
Easy with 50% error Erdős
Bad news: Max Cut is NP-hard
Approximations?
How to find the minimum energy position?
dim=1: Min = 4·Max Cut Min 4·Max Cut in any dim
dim=2: probably hard
dim=n: Polynomial time solvable!semidefinite optimization
spring (repulsive)
2( )i jij E
u u
EEnergy:
| | 1iu
random hyperplane
Expected number of edges cut:
2angle( , ) / .22 | | .22 | min | .88MaxCuti j i jij E ij E
u u u u
E|
Probability of edge ij cut: angle( , ) /i ju u
minimum energyin n dimension
Stresses of tensegrity frameworks
bars
struts
cables x y( )ijS x y
( ) 0ij j ij
S x x Equilibrium:
There is no non-zero stress on the edges of a convex polytope
Cauchy
Every simplicial 3-polytope is rigid.
If the edges of a planar graph are colored red and bule,then (at least 2) nodes where the colors are
consecutive.
3
1
4
5
3
9
10
10
9
2
2
2
3
3
Every triangulation of a quadrilateral can be
represented by a square tiling of a rectangle.
Schramm
Coin representation
Every planar graph can be represented by touching circles
Koebe (1936)
Discrete Riemann Mapping Theorem
Representation by orthogonal circles
A planar triangulation can be represented byorthogonal circles
no separating 3- or 4-cycles Andre’ev
Thurston
Polyhedral version
Andre’ev
Every 3-connected planar graph
is the skeleton of a convex polytope
such that every edge
touches the unit sphere
From polyhedra to circles
horizon
From polyhedra to representation of the dual