Which graphs are extremal?

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# Which graphs are extremal? - PowerPoint PPT Presentation

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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Which graphs are extremal?

LászlóLovász

Eötvös Loránd University

Budapest

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

Some old and new results from extremal graph theory

Probability that random map

V(F)V(G)preserves edges

Some old and new results from extremal graph theory

Theorem (Goodman):

t( ,G) – 2t( ,G) + t( ,G) ≥ 0

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

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)

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

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

• 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?

• 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?

Which inequalities between densities are valid?

IfvalidforlargeG,

thenvalidforall

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

The main trick in the proof

1

t( ,G) – 2t( ,G) + t( ,G) = 0

0

1/2

2/3

3/4

1

Which inequalities between densities are valid?

Undecidable…

Hatami-Norine

…but decidable with an arbitrarily small error.

L-Szegedy

• 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?

Computing with graphs

-2 +  0

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

-  0

Kruskal-Katona:

Goodman:

Erdős:

-  0

Computing with graphs

2

-

-

+

=

-

-

+

2

-

+

=

-2

2

2

-

-

-

+

=

-

-

+2

+

-4

+2

+2

- 2

+

Goodman’s Theorem

-2 +  0

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

A weak Positivstellensatz

(ignoring labels

and isolated nodes)

L - Szegedy

• 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?

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

Limit objects

(graphons)

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

Limit objects

(graphons)

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

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

Borgs-Chayes-L-Sós-Vesztergombi

Example: graph limit

A random graphwith 100

nodes and with 2500 edges

Example: graph limit

A randomly grown uniform

attachment graphon200 nodes

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

Connection matrices

...

M(f, k)

k=2:

...

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

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

Is there always an extremal graph?

No, but there is always an extremal graphon.

The space of graphons

is compact.

• 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?

Extremal graphon problem

Given quantum graphs g0,g1,…,gm,

find max t(g0,W)

subject to t(g1,W) = 0

t(gm,W) = 0

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 ??

Finitely forcible graphons

Graham-

Chung-

Wilson

1/2

Goodman

Which graphs are extremal?

Stepfunction:

Stepfunctions  finite graphs

with node and edgeweights

Stepfunctions are finitely forcible

L – V.T.Sós

Which graphons are finitely forcible?

Is the following graphon finitely forcible?

angle <π/2

Thanks, that’s

all for today!

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

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

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

The local version

Let

Then t(F,W)  1.

The idea of the proof

Main Lemma:

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

then t(F,U) ≤ t(C2r,U).

Common graphs

?

Erdős:

Thomason

Common graphs

F common:

Common graphs:

Sidorenko graphs

(bipartite?)

Non-common graphs:

 graph containing

Jagger, Stovícek, Thomason

Common graphs

F common:

is common. Franek-Rödl

8 +2 + +4

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

Common graphs

F locally common:

is locally common. Franek-Rödl

12 +3 +3 +12 +

12 2 +3 2 +3 4 +12 4 + 6

Common graphs

graph containing is locally common.

graph containing is locally common

but not common.

Not locally common:

Common graphs

F common:

is common. Franek-Rödl

 - 1/21/2  - 1/21/2

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

Common graphs

F common:

Common graphs:

Sidorenko graphs

(bipartite?)

Non-common graphs:

 graph containing

Jagger, Stovícek, Thomason

Some old and new results from extremal graph theory

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

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

Finitely expressible properties

d-regular

d-regular graphon:

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.