January 2016

Spectra of graphs and geometric representations

László Lovász Hungarian Academy of Sciences

Eötvös Loránd Universitylovasz@cs.elte.hu

January 2016

Happy Birthday, Noga!

January 2016

Extreme graphs?Shannon capacity?Strong regularity lemma?Property testing?Combinatorial Nullstellensatz?Anti-Hadamard matrices?Optimization?Eigenvalues?Eigenvalues!

January 2016

The eigenvalue gap

Laplacian:

1

2

0 ... 1 0 10 ... 0 1 0

1 00 1

1 0 n

dd

L

d

M M

O

adjacent positions

degrees

Eigenvalues: 1 2 ... n

January 2016

Graphs and the eigenvalue gap

Gap between 1 and 2 expander graph

Alon - MilmanAlon

1 < 2 graph is connected

January 2016

G-matrix: , symmetric, 0 ( , )

V V

ij

M MM ij E i j

¡

G = (V,E): simple graph, V=[n]

well-signed G-matrix: 0 ( )ijM ij E

Graphs, matrices, geometric representations

Want to understand: UM=0, M: G-matrix, Udxn

d=rank(U)=corank(M)

really good G-matrix: well-signed, one negative eigenvalue

January 2016

UM = 0

nullspace representation

M U: nullspace representation

unique up to linear transformation

cycle fixed toconvex polygon

edges replaced byrubber bands

MU: rubber bands

G is 3-connected planar, fixed cycle a face

planar embedding

Tutte

MU: rubber bands

2( )ij i jij E

M u u

EEnergy:

MU: rubber bands

0ij ij jj i

iij

uM M M

Equilibrium:( )

( ) 0ij j ij N i

M u u

(j free node)

stress matrix

stress in rubber bandorstrength of rubber band

January 2016

MU: rubber bands

Mij: stress

define stress Mij so that

equilibrium condition

holds at all nodes

January 2016

UM: bar-and-joint structures

--+

+ + +

0

0

ij jj

ijj

M u

M

M has corank 3 and is positive semidefinite.

Connelly

January 2016

UM: bar-and-joint structures

--+

+ + +

2

,

2

,

( )

( ) ( ) ( )

ij i ji j

ij i i j ji j

u M u u

u x M u x u x u x

E

E E E

ui

Mij

January 2016

Braced stresses

UM = 0

nullspace representation

M’

MU 0

U’U’M’=0

January 2016

Braced stresses

P P*

( )uvMp q u v

u

v

q

p

January 2016

UM: canonical stress on 3-polytopes

Canonical braced stress

P P*

u

v

q

p

January 2016

UM: canonical stress on 3-polytopes

The canonical braced stress matrixhas 1 negative and 3 zero eigenvalues. L

(really good G-matrix)

January 2016

MU: the Colin de Verdière number

G: connected graph

Roughly: multiplicity of second largest eigenvalue

of adjacency matrix

And: non-degeneracy condition on weightings

Largest has multiplicity 1.

But: maximize over weighting the edges and diagonal entries

Mii arbitrary

Strong Arnold Property( ) maxcorank( )G M

normalization

M=(Mij): well-signed G-matrix•

M has =1 negative eigenvalue•

January 2016

[(G)-connected]

μ(G) is minor monotone

deleting and contracting edges

μ k 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

January 2016

Colin de Verdière number Basic properties

μ(G)1 G is a path

μ(G)3 G is a planar

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

μ(G)2 G is outerplanar

January 2016

Colin de Verdière number Special values

0x 0x 0x

supp ( ), supp ( )xx are connected.

discrete Courant Nodal TheoremJanuary 2016

M: really good G-matrix

Mx = 0

supp(x) minimal

Van der Holst’s lemma

like convex polytopes?

or…

connected

January 2016

Van der Holst’s lemma for nullspace representation

S+

S-

Corank bound

January 2016

January 2016

The eigenvalue gap

Gap between 1 and 2 expander graph

Alon - MilmanAlon

1 < 2 graph is connected

2 < 3 G[supp+(v2)], G[supp-(v2)] are connected

van der Holst

January 2016

The eigenvalue gap

Gap between 2 < 3 G[supp+(v2)], G[supp-(v2)]

are expanders

expander expander

?

Use (v2)i2 as weights!

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

LJanuary 2016

MU: Steinitz representations

μ(G)1 G is a path

μ(G) 3 G is a planar

μ(G)2 G is outerplanar

μ(G)4 G is linklessly embeddable in 3-spaceL - Schrijver

January 2016

Colin de Verdière number Special values

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 nullspace representation gives

outerplanar outerplanar embedding in 2

G 3-connected nullspace representation gives

planar planar embedding in 2, and also

Steinitz representation

L-Schrijver; L

January 2016

January 2016

Computing G-matrices

Input: A 2-connected graph G=(V,E).

Output: Either an outerplanar embedding of G,

or a really good G-matrix with corank 3.

Special case: G 3-connected planar

Steinitz representation of G

January 2016

UM: circulations

h: circulation on edges ij with ui and uj not parallel

i ui 2

, ( , ) 0ij j i ij j i ijj i j i j i

area M u u M area u u h

( , ) ( , ),area( , )

arbitrary ( , )

i ji j

i j

ij

h i j ij E u uu uM

ij E u u

P

P

ij j i ij j ij i j i

iiM u u MM u u

P Every G-matrix arises this way

January 2016

M well-signed h is a counterclockwise circulation

M has one negative eigenvalue ?

UM: circulations

January 2016

Shifting the origin

ui: nullspace representation, |ui|=1

M: really good G-matrix with corank 2

January 2016

Many more nice theorems,Noga!