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Which graphs are extremal?. L á szl ó Lov á sz Eötvös Loránd University Budapest . Some old and new results from extremal graph theory. Extremal:. Theorem (Goodman):. Tur á n’s Theorem (special case proved by Mantel): G contains no triangles  #edges n 2 /4.

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slide1

Which graphs are extremal?

LászlóLovász

Eötvös Loránd University

Budapest

slide2

Some old and new results from extremal graph theory

Extremal:

Theorem (Goodman):

Turán’s Theorem (special case proved by Mantel):

G contains no triangles  #edgesn2/4

slide3

Some old and new results from extremal graph theory

Probability that random map

V(F)V(G)preserves edges

Homomorphism: adjacency-preserving map

slide4

Some old and new results from extremal graph theory

Theorem (Goodman):

t( ,G) – 2t( ,G) + t( ,G) ≥ 0

t( ,G) = t( ,G)2

slide5

Some old and new results from extremal graph theory

n

k

Kruskal-Katona Theorem (very special case):

t( ,G)2≥ t( ,G)3

t( ,G) ≥ t( ,G)

slide6

Some old and new results from extremal graph theory

Kruskal-Katona

Razborov 2006

Fisher

Goodman

Bollobás

Mantel-Turán

Lovász-Simonovits

Semidefiniteness and extremal graph theory

Tricky examples

1

0

1/2

2/3

3/4

1

slide7

Some old and new results from extremal graph theory

Theorem (Erdős):

G contains no 4-cycles  #edgesn3/2/2

(Extremal: conjugacy graph of finite

projective planes)

t( ,G) ≥ t( ,G)4

slide8

General questions about extremal graphs

  • Which inequalities between subgraph densities
  • are valid?

- Can all valid inequalities be proved using

just Cauchy-Schwarz?

- Is there always an extremal graph?

- Which graphs are extremal?

slide9

General questions about extremal graphs

  • Which inequalities between subgraph densities
  • are valid?

- Can all valid inequalities be proved using

just Cauchy-Schwarz?

- Is there always an extremal graph?

- Which graphs are extremal?

slide10

Which inequalities between densities are valid?

IfvalidforlargeG,

thenvalidforall

slide11

Analogy with polynomials

p(x1,...,xn)0

for all x1,...,xnRdecidable Tarski

  • p = r12 + ...+rm2 (r1, ...,rm:rational functions)
  • „Positivstellensatz”Artin

for all x1,...,xnZundecidable Matiyasevich

slide13

The main trick in the proof

1

t( ,G) – 2t( ,G) + t( ,G) = 0

0

1/2

2/3

3/4

1

slide14

Which inequalities between densities are valid?

Undecidable…

Hatami-Norine

…but decidable with an arbitrarily small error.

L-Szegedy

slide15

General questions about extremal graphs

  • Which inequalities between subgraph densities
  • are valid?

- Can all valid inequalities be proved using

just Cauchy-Schwarz?

- Is there always an extremal graph?

- Which graphs are extremal?

slide16

Computing with graphs

-2 +  0

Writea≥ 0if t(a,G) ≥ 0for every graph G.

-  0

Kruskal-Katona:

Goodman:

Erdős:

-  0

slide17

Computing with graphs

2

-

-

+

=

-

-

+

2

-

+

=

-2

2

2

-

-

-

+

=

-

-

+2

+

-4

+2

+2

- 2

+

Goodman’s Theorem

-2 +  0

slide18

Positivstellensatz for graphs?

If a quantum graph x is sum of squares (ignoring

labels and isolated nodes), then x ≥ 0.

Question: Suppose thatx ≥ 0. Does it follow that

No!

Hatami-Norine

slide19

A weak Positivstellensatz

(ignoring labels

and isolated nodes)

L - Szegedy

slide20

General questions about extremal graphs

  • Which inequalities between subgraph densities
  • are valid?

- Can all valid inequalities be proved using

just Cauchy-Schwarz?

- Is there always an extremal graph?

- Which graphs are extremal?

slide21

Is there always an extremal graph?

Minimize over x0

always >1/16,

arbitrarily close for random graphs

Real numbers are useful

minimum is not attained

in rationals

Minimize t(C4,G) over graphs

with edge-density 1/2

Quasirandom graphs

Graph limits are useful

minimum is not attained

among graphs

slide22

Limit objects

(graphons)

slide23

Graphs  Graphons

0 0 1 0 0 1 1 0 0 0 1 0 0 1

0 0 1 0 1 0 1 0 0 0 0 0 1 0

1 1 0 1 0 1 1 1 1 0 1 0 1 1

0 0 1 0 1 0 1 0 1 0 1 1 0 0

0 1 0 1 0 1 1 0 0 0 1 0 0 1

1 0 1 0 1 0 1 1 0 1 1 1 0 1

1 1 1 1 1 1 0 1 0 1 1 1 1 0

0 0 1 0 0 1 1 0 1 0 1 0 1 1

0 0 1 1 0 0 0 1 1 1 0 1 0 0

0 0 0 0 0 1 1 0 1 0 1 0 1 0

1 0 1 1 1 1 1 1 0 1 0 1 1 1

0 0 0 1 0 1 1 0 1 0 1 0 1 0

0 1 1 0 0 0 1 1 0 1 1 1 0 1

1 0 1 0 1 1 0 1 0 0 1 0 1 0

G

AG

WG

slide24

Limit objects

(graphons)

t(F,WG)=t(F,G)

(G1,G2,…) convergent: Ft(F,Gn) converges

Borgs-Chayes-L-Sós-Vesztergombi

slide25

Example: graph limit

A random graphwith 100

nodes and with 2500 edges

slide26

Example: graph limit

A randomly grown uniform

attachment graphon200 nodes

slide27

Limit objects: themath

For every convergent graph sequence (Gn)

there is a WW0 such thatGnW.

Conversely, W(Gn) such thatGnW.

L-Szegedy

W is essentially unique

(up to measure-preserving transformation).

Borgs-Chayes-L

slide28

Connection matrices

...

M(f, k)

k=2:

...

slide29

Semidefinite connection matrices

f: graph parameter

W: f = t(.,W)

k M(f,k) is positive semidefinite,

f()=1 and f is multiplicative

L-Szegedy

slide30

Proof of the weak Positivstellensatz (sketch2)

the optimum of a semidefinite program is 0:

minimize

subject to M(x,k)positive semidefinite k

x(K0)=1

x(GK1)=x(G)

Apply Duality Theorem of semidefinite programming

slide31

Is there always an extremal graph?

No, but there is always an extremal graphon.

The space of graphons

is compact.

slide32

General questions about extremal graphs

  • Which inequalities between subgraph densities
  • are valid?

- Can all valid inequalities be proved using

just Cauchy-Schwarz?

- Is there always an extremal graph?

- Which graphs are extremal?

slide33

Extremal graphon problem

Given quantum graphs g0,g1,…,gm,

find max t(g0,W)

subject to t(g1,W) = 0

t(gm,W) = 0

slide34

Finitely forcible graphons

Every finitely forcible graphon is extremal:

minimize

Finite forcing

GraphonW is finitely forcible:

Every unique extremal graphon is finitely forcible.

??Every extremal graph problem has a

finitely forcible extremal graphon ??

slide35

Finitely forcible graphons

Graham-

Chung-

Wilson

1/2

Goodman

slide36

Which graphs are extremal?

Stepfunction:

Stepfunctions  finite graphs

with node and edgeweights

Stepfunctions are finitely forcible

L – V.T.Sós

slide38

Which graphons are finitely forcible?

Is the following graphon finitely forcible?

angle <π/2

slide39

Thanks, that’s

all for today!

slide40

The Simonovits-Sidorenko Conjecture

?

F bipartite, G arbitrary t(F,G) ≥ t(K2,G)|E(F)|

Known when F is a tree, cycle, complete bipartite…

Sidorenko

F is hypercube

Hatami

F has a node connected to all nodes

in the other color class

Conlon,Fox,Sudakov

F is "composable"

Li, Szegedy

slide41

The Simonovits-Sidorenko Conjecture

Two extremal problems in one:

For fixed G and |E(F)|, t(F,G) is minimized

by F=

For fixed F and t( ,G), t(F,G) is minimized

by random G

asymptotically

slide42

The integral version

Let WW0, W≥0, ∫W=1. Let F be bipartite.

Then t(F,W)≥1.

?

For fixed F, t(F,W) is minimized over W≥0, ∫W=1

by W1

slide43

The local version

Let

Then t(F,W)  1.

slide45

The idea of the proof

Main Lemma:

If -1≤ U ≤ 1, shortest cycle in F is C2r,

then t(F,U) ≤ t(C2r,U).

slide46

Common graphs

?

Erdős:

Thomason

slide47

Common graphs

Hatami, Hladky, Kral, Norine, Razborov

F common:

Common graphs:

Sidorenko graphs

(bipartite?)

Non-common graphs:

 graph containing

Jagger, Stovícek, Thomason

slide49

Common graphs

F common:

is common. Franek-Rödl

8 +2 + +4

= 4 +2 +( +2 )2 +4( - )

slide50

Common graphs

F locally common:

is locally common. Franek-Rödl

12 +3 +3 +12 +

12 2 +3 2 +3 4 +12 4 + 6

slide51

Common graphs

graph containing is locally common.

graph containing is locally common

but not common.

Not locally common:

slide52

Common graphs

F common:

is common. Franek-Rödl

 - 1/21/2  - 1/21/2

8 +2 + +4 = 4 +2 +( -2 )2

slide53

Common graphs

Hatami, Hladky, Kral, Norine, Razborov

F common:

Common graphs:

Sidorenko graphs

(bipartite?)

Non-common graphs:

 graph containing

Jagger, Stovícek, Thomason

slide54

Some old and new results from extremal graph theory

Theorem (Erdős-Stone-Simonovits):(F)=3

slide55

Computing with graphs

1

2

Graph parameter: isomorphism-invariant function

on finite graphs

k-labeled graph: k nodes labeled 1,...,k,

any number of unlabeled nodes

slide56

Finitely expressible properties

d-regular

d-regular graphon:

slide57

Finitely expressible properties

W is 0-1 valued,

and can be rearranged

to be monotone decreasing

in both variables

W is 0-1 valued

"W is 0-1 valued" is not finitely expressible in terms of

simple gaphs.