Eigenvalues and
geometric representations
of graphs
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA 98052
lovasz@microsoft.com
planar graph
Every planar graph can be drawn
in the plane with straight edges
Fáry-Wagner
3-connected planar graph
Every 3-connected planar graph
is the skeleton of a convex 3-polytope.
Steinitz 1922
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 to
convex polygon
edges replaced by
rubber bands
Energy: E 
2
(
u

u
)
 i j
ij E
Equilibrium:
1
ui 
di

j N ( i )
uj
Demo
G 3-connected planar
rubber band embedding is planar
Tutte
(Easily) polynomial time
computable
Lifts to Steinitz
representation if
outer face is a triangle
Maxwell-Cremona
If the outer face is a triangle, then the Tutte representation
is a projection of a Steinitz representation.
p
 vF 
 1 
F’
F
p’
u p - u p ' = (vF - vF ' )¿
If the outer face is a not a triangle, then go to the polar.
P
d
: polytope,
P*  { y 
d
0 P
: xT y  1 x  P}: polar polytope
Rubber bands and connectivity
G: arbitrary graph
A,B  V, |A|=|B|=3
A: triangle
edges: rubber bands
Energy: E 
 c (u
ij
i
uj)
2
ij E
Equilibrium:
ui 

j N ( i )
cij
di
uj
di 

j N ( i )
cij
For almost all choices of edge strengths:
B noncollinear

3 disjoint (A,B)-paths
()
cutset
Linial-LWigderson
For almost all choices of edge strengths:
B affine indep

k disjoint (A,B)-paths
()
strengthen
Linial-LWigderson
 edges strength s.t. B is independent
no algebraic relation
between edge strength
for a.a. edge strength, B is independent
The maximum cut problem
maximize
Applications: optimization, statistical mechanics…
Bad news: Max Cut is NP-hard
Approximations?
Easy with 50% error
C
Erdős
Bad news: Max Cut is NP-hard
Approximations?
Easy with 50% error
C
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?
Easy with 50% error
Erdős
NP-hard with 6% error
Hastad
Polynomial with 12% error
Goemans-Williamson
spring (repulsive)
Energy: E  
2
(
u

u
)
 i j
ij E
| ui | 1
How to find the minimum energy position?
dim=1: Min E = 4·Max Cut
Min E  4·Max Cut in any dim
dim=2: probably hard

dim=n: Polynomial time solvable!
semidefinite
optimization
minimum energy
in n dimension
random hyperplane
Probability of edge ij cut: angle(ui , u j ) / 
Expected number of edges cut:
 angle(ui , u j ) /   .22
ij E
2
|
u

u
|
 i j  .22 | min E||  .88Max Cut
ij E
Stresses of tensegrity frameworks
bars
struts
x
cables
Equilibrium:
S
j
ij
Sij ( x  y )
( x j  xi )  0
y
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.
10
3
9
3
3
2
4
1
9
2
5
3
2
10
Every triangulation of a quadrilateral can be
represented by a square tiling of a rectangle.
Schramm
Coin representation Koebe (1936)
Every planar graph can be represented by touching circles
Discrete Riemann Mapping Theorem
Representation by orthogonal circles
A planar triangulation can be represented by
orthogonal circles

no separating 3- or 4-cycles
Andre’ev
Thurston
Polyhedral version
Every 3-connected planar graph
is the skeleton of a convex polytope
such that every edge
touches the unit sphere
Andre’ev
From polyhedra to circles
horizon
From polyhedra to representation of the dual