graph algebras and extremal graph theory
DESCRIPTION
Graph algebras and extremal graph theory. L á szl ó Lov á sz. Three questions in 2012. Mike Freedman : Which graph parameters are partition functions ?. Jennifer Chayes : What is the limit of a randomly growing graph sequence (like the internet)?. - PowerPoint PPT PresentationTRANSCRIPT
Graph algebras andextremal graph theory
László Lovász
April 2013 1
April 2013 2
April 2013 3
Three questions in 2012
Mike Freedman: Which graph parametersare partition functions?
Jennifer Chayes: What is the limit of a randomlygrowing graph sequence (like the internet)?
Vera Sós: How to characterize generalizedquasirandom graphs?
April 2013 4
Three questions in 2012
Mike Freedman: Which graph parametersare partition functions?
Jennifer Chayes: What is the limit of a randomlygrowing graph sequence (like the internet)?
Vera Sós: How to characterize generalizedquasirandom graphs?
April 2013
Graph parameter: isomorphism-invariant function on finite graphs
5
Which parameters are homomorphism functions?
hom( , ) : # of homomorphisms of intoFF GG Homomorphism: adjacency-preserving map
Partition functions of vertex coloring models
April 2013
| ( )|
hom( , )| ( ) |
( , ) V F
F GV G
t F G Probability that random
mapV(F)V(G) is a hom
6
Weighted version:
: , :( , , , ) , VH V E E ¡ ¡
( )(: ( ) (
( )( )))
)(
hom( , ) :
i jij E
ii V F FV F V H
F H
Which parameters are homomorphism functions?
April 2013
1 2
1 2
1 2 ,
, :
:
-labeled graphs
labeled nodes identified
k
G G
G G
GG È=
7
k-labeled graph: k nodes labeled 1,...,k, any number of unlabeled nodes
1
2
Which parameters are homomorphism functions?
April 2013
k=2:
...
...
( )f
M(f, k)
8
Connection matrices
April 2013
f = hom(.,H) for some weighted graph H on q nodes
(k) M(f,k) is positive semidefinite, rank qk
Freedman - L - Schrijver
Which parameters are homomorphism functions?
9
Difficult direction:
Easy but useful direction:
April 2013
Related (?) characterizations
10
f=t(.,W) for some ``graphon’’ (graph limit) W
L-Szegedy
f=t(.,h) for some ``edge coloring model’’ W
Szegedy
f=hom(.,H) for some edge-weighted graph H
Schrijver
f=hom(.,G) for some simple graph G
L-Schrijver
tensor algebras
Schrijver
dual problem,
categories
L-Schrijver
April 2013
Limit objects
11
(graphons)
20 : [0,1] [0,1] symmetric, measurableW W
G
0 0 1 0 0 1 1 0 0 0 1 0 0 10 0 1 0 1 0 1 0 0 0 0 0 1 01 1 0 1 0 1 1 1 1 0 1 0 1 10 0 1 0 1 0 1 0 1 0 1 1 0 00 1 0 1 0 1 1 0 0 0 1 0 0 11 0 1 0 1 0 1 1 0 1 1 1 0 11 1 1 1 1 1 0 1 0 1 1 1 1 00 0 1 0 0 1 1 0 1 0 1 0 1 10 0 1 1 0 0 0 1 1 1 0 1 0 00 0 0 0 0 1 1 0 1 0 1 0 1 01 0 1 1 1 1 1 1 0 1 0 1 1 10 0 0 1 0 1 1 0 1 0 1 0 1 00 1 1 0 0 0 1 1 0 1 1 1 0 11 0 1 0 1 1 0 1 0 0 1 0 1 0 AG
WG
Graphs Graphons
April 2013 12
April 2013
( ) ( )[0,1]
( , )( , )V F
i jij E F
W x x dxt F W
Limit objects
13
(graphons)
simpl: e ( , ) ( , )nn F tW FG G t F W
20 : [0,1] [0,1] symmetric, measurableW W
t(F,WG)=t(F,G)
(G1,G2,…) convergent: simple F t(F,Gn) converges
Borgs-Chayes-L-Sós-Vesztergombi
For every convergent graph sequence (Gn)
there is a graphon W such that GnW.For every graphon W there is a graph sequence (Gn) such that GnW.
April 2013 14
Limit objects
L-Szegedy
W is essentially unique (up to measure-preserving transformation).
Borgs-Chayes-L
f=t(.,W) for some graphon W f is multiplicative, reflection positive, and f(K1)=1.
L-Szegedy
April 2013 15
April 2013
k-labeled quantum graph:
GG
x x Gfinite formal sum ofk-labeled graphs
1
2
infinite dimensional linear spaces
16
Computing with graphs
Gk = {k-labeled quantum graphs}
infinite formal sum with G |xG|
Lk = {k-labeled quantum graphs (infinite sums)}
April 2013
kG is a commutative algebra with unit element ...
Define products:
1 2
1 2
1 2,
G G GG GG
GG
x y Gx G GG yæ öæ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç÷ ÷è øè ø=å å å
17
Computing with graphs
G1,G2: k-labeled graphsG1G2 = G1G2, labeled nodes identified
April 2013
Inner product:
(, )' 'f
fG GGG
f: graph parameter
, ,f f
x yz xy z
( )GGG G
x Gf x G f
:,f
x y extend linearly
18
Computing with graphs
April 2013
2
0
0
,
( )
( , ) is positive semidefinite
kf
k
x x x
f x x
M f k
G
G
f is reflection positive
Computing with graphs
19
April 2013
//k k ff G KG{ : , 0 }k kff x x y yG GK = "= Î Î
Factor out the kernel:
20
Computing with graphs
/ has finitedimension
( , ) has finite rank
k f
M f k
G
1
2 0
has a basis ,..., such that
,
kk q
i i i j
p p
p p p p
G
1 ... k kqp p u
In particular, for k=1:
1 1... qp p u
Computing with graphs
April 2013 21
Assume that rk(M(f,k))=qk.
ip
jp2i jp p G
Computing with graphs
April 2013 22
1
1
2,
( ) { ,... },
,q
i
i ji j
i
ij
V H p p
p k
k p p
pipj form a basis of G2.
Assume that rk(M(f,k))=qk.
April 2013 23
April 2013
Turán’s Theorem (special case proved by Mantel):G contains no triangles #edgesn2/4
Theorem (Goodman): t(K3,G) (2t(K2,G) -1) t(K2,G)
Extremal:
24
Some old and new results from extremal graph theory
April 2013
Theorem (Erdős):G contains no 4-cycles #edgesn3/2/2
(Extremal: conjugacy graph of finite projective planes)
25
Some old and new results from extremal graph theory
t(C4,G) t(K2,G)4
April 2013
k=2:
26
Connection matrices
t( ,G) t( ,G)2 t( ,G)4
April 2013 27
Which inequalities between densities are valid?
Undecidable…
Hatami-Norine
1
( , ): 0? ?m
ii iG t F G
If valid for large G,
then valid for all
…but decidable with an arbitrarily small error.
L-Szegedy
April 2013
Write x ≥ 0 if hom(x,G) ≥ 0 for every graph G.
Turán: -2 + 0³
Kruskal-Katona: - 0³
Blakley-Roy: - 0³
Computing with graphs
28
April 2013
- +-2
= - +-
- +- 2+2 2- = - +- +2 -4 +2
Goodman’s Theorem
Computing with graphs
29
+- 2+- 2 ≥ 0
2- = 2 -4 +2
t( ,G) – 2t( ,G) + t( ,G) ≥ 0
April 2013
2 221 1
2( ... . .) ?. mn xz y yz
Question: Suppose that x ≥ 0. Does it follow that2 21 .. ?. mx y y
Positivstellensatz for graphs?
30
No! Hatami-Norine
If a quantum graph x is a sum of squares (ignoring labels and isolated nodes), then x ≥ 0.
April 2013
Let x be a quantum graph. Then x 0
2 2
1 1 10 ,..., ...m k mk y y G x y y
A weak Positivstellensatz for graphs
31
L-Szegedy
April 2013
An exact Positivstellensatz for graphs
32
F*: graphs partially labeled by different
nonnegative integers
L: functions f: F* such that F |f(F)|<.
1 2
1 1 2 2 11 2 2,
( ) ( ) ( ) ( )G GG G
f f G f G GG G f G G Gæ öæ ö÷ ÷ç ç÷ ÷ç ç÷ ÷ç ç÷ ÷è øè ø=å å å
( )G
f f G G« å
April 2013
x 0
2
1 21
, ,... ii
y y x y
L
An exact Positivstellensatz for graphs
33
the optimum of a semidefinite program is 0:
minimize
subject to M(f,k) positive semidefinite for all k
f(K1)=1
f(GK1)=f(G)
April 2013
Proof of the weak Positivstellensatz (sketch2)
Apply Duality Theorem of semidefinite programming
34
0: ( , )i iG t F G
( )i if F
April 2013 35
Pentagon density in triangle-free graphs
5
51 2( , ) 105 625
t C G
April 2013 36
3 52( , ) 0 ( , )
625t K G t C G
Conjecture: Erdős 1984
Proof: Hatami-Hladky-Král-Norine-Razborov 2011
Pentagon density in triangle-free graphs
1 2 3 4
2
2
2625 25
1097 34158266 ... 2400 25
|Q Q Q Q
o
o
April 2013 37
1
1 2 3 2
3
13900 -671 -12807-671 31334 -51136-12807 -51136 98157
zz z z z
z
3-labeledquantum graphs
positive semidefinite
Pentagon density in triangle-free graphs
April 2013 38
April 2013 39