Eigenvalues and

geometric representations

of graphs

László Lovász

Microsoft Research

One Microsoft Way, Redmond, WA 98052

lovasz@microsoft.com

Steinitz 1922

Every 3-connected planar graphis the skeleton of a convex 3-polytope.

3-connected planar graph

Representation by special polyhedra

Every 3-connected planar graph

is the skeleton of a convex polytope

such that every edge

touches the unit sphere

Koebe-Andreev-Thurston

From polyhedra to circles

horizon

From polyhedra to the polar

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 Andreev

Thurston

/ 2ija

The Colin de Verdière number

G: connected graph

Roughly: multiplicity of second largest eigenvalue

of adjacency matrix

But: non-degeneracy condition on weightings

Largest has multiplicity 1.

But: maximize over weighting the edges and diagonal entries

Mii arbitrary

Strong Arnold Property

( ) max corank ( )G M

normalization

M=(Mij): symmetric VxV matrix•

Mij

<0, if ijE

0, if ,ij E i j •

M has =1 negative eigenvalue•

( )ijX X symmetric, 0 for andijX ij E i j

00MX X •

The Colin de Verdière number of a graph

Basic Properties

μ(G) is minor monotone

deleting and contracting edges

μk is polynomial timedecidable for fixed k

for μ>2, μ(G) is invariant under subdivision

for μ>3, μ(G) is invariant under Δ-Y transformation

μ(G)1 G is a path

Special values

μ(G)3 G is a planar

Colin de Verdière, using pde’sVan der Holst, elementary proof

μ(G)2 G is outerplanar

0x 0x 0x

supp ( ), supp ( )xx are connected.

Van der Holst’s lemma

Courant’s Nodal Theorem

0Mx

supp( ) minimalx

0

0 0

0

0

0

_

_

+

+

_0

+

_

G planar corank of M is at most 3.

representation of G in μ

Nullspace representation:

0ij jj

M u

1

2

n

u

u

u

11 12 1

21 22 2

1 2

...

...

...n n n

x x x

x x x

x x x

basis of nullspace of M1 2 .. :.x x x

corank of M is at most 3 G planar .

Van der Holst’s Lemma, geometric form

like convex polytopes?

or…

connected

G 3-connected planar

nullspace representation,scaled to unit vectors,gives embedding in S2

L-Schrijver

G 3-connected planar

nullspace representationcan be scaled to convex polytope L

nullspace representationplanar embedding

P P*

Colin de Verdière matrix M

Steinitz representationP

( )uvMp q u v

u

v

q

p

μ(G)1 G is a path

Special values

μ(G)3 G is a planar

Colin de Verdière, using pde’sVan der Holst, elementary proof

μ(G)2 G is outerplanar

μ(G)4 G is linklessly embeddable in 3-space

L - Schrijver

Linklessly embedable graphs

homological, homotopical,…equivalent

embedable in 3 without linked cycles

Apex graph

G linklessly embedable

G has no minor in the “Petersen family”

Robertson – Seymour - Thomas

G 4-connected

linkless embed. nullspace representation gives

linkless embedding in 3

?

G path nullspace representation gives

embedding in 1

properly normalized

G 2-connected

outerplanar nullspace representation gives

outerplanar embedding in 2

G 3-connected

planar nullspace representation gives

planar embedding in 2, and also

Steinitz representationL-Schrijver; L

μ(G)1 G is a path

…

μ(G)n-4 complement G is planar_

~

Kotlov-L-Vempala

Special values

μ(G)3 G is a planar

Colin de Verdière, using pde’sVan der Holst, elementary proof

μ(G)2 G is outerplanar

μ(G)4 G is linklessly embeddable in 3-space

L - SchrijverKoebe-Andreevrepresentation

The Gram representation

1 1 1 1( )TA Q M Q Q MQ J pos semidefinite

, : diag( )M Q

1, ( );

1, ( ).Ti j

ij E Gu u

ij E G

if

if

Kotlov – L - Vempala

1( )T nij i j iA u u u

Gram representation

Properties of the Gram representation

ui is a vertex of P

1: conv( ,..., )nP u u | | 1iu exceptional

Assume: G has no twin nodes, and | | 1iu

( ) i juij E G u is an edge of P

0 int P

If G has no twin nodes, and μ(G)n-4, then

is planar.G