Eigenvalues and geometric representations of graphs László Lovász Microsoft Research One...

Eigenvalues and

geometric representations

of graphs

László Lovász

Microsoft Research

One Microsoft Way, Redmond, WA 98052

[email protected]

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



Approximations?



Approximations?

NP-hard with 6% error Hastad

Polynomial with 12% error

Goemans-Williamson



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

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

Eigenvalues and geometric representations of graphs László Lovász Microsoft Research One...

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