1. Graph algebras and graph limits László Lovász
Eötvös Loránd University, Budapest
Joint work with Christian Borgs, Jennifer Chayes,
Balázs Szegedy, Vera Sós and Katalin Vesztergombi
July 2010

2. Some old and new results from extremal graph theory
Turán’s Theorem (special case proved by Mantel):
G contains no triangles  #edgesn2/4
Extremal:
Theorem (Goodman):
July 2010

3. Some old and new results from extremal graph theory
Kruskal-Katona Theorem (very special case): n k
July 2010

4. Some old and new results from extremal graph theory
Semidefiniteness and extremal graph theory
Tricky examples
Kruskal-Katona 1
Goodman
Razborov 2006
Fisher
Bollobás
Lovász-Simonovits
1/2
2/3
3/4 1
Mantel-Turán
July 2010

5. 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)
Theorem (Chung-Graham-Wilson):
Quasirandom graphs
July 2010

6. General questions about extremal graphs
Which inequalities between subgraph densities
are valid?
- Is there always an extremal graph?
- Which graphs are extremal?
July 2010

7. Homomorphism functions
Homomorphism: adjacency-preserving map
coloring
independent
set
triangles
July 2010

8. Homomorphism functions
Probability that random map
V(G)V(H) is a hom
Weighted version:
July 2010

9. Homomorphism functions
Examples:
hom(G, ) = # of independent sets in G
if G has no loops
July 2010

10. Homomorphism functions3 3 -1
1/4
1/4 -1 -1 -1 -1
1/4
1/4 -1 3 3 H 1 2 H
partition functions in statistical physics...
July 2010

11. Which parameters are homomorphism functions?
Graph parameter: isomorphism-invariant function on finite graphs 1 2
k-labeled graph: k nodes labeled 1,...,k,
any number of unlabeled nodes
1,3 2
k-multilabeled graph: nodes labeled 1,...,k,
any number of unlabeled nodes
July 2010

12. Connection matrices
...
M(f, k)
k=2:
...
July 2010

13. Which parameters are homomorphism functions?
is positive semidefinite
and has rank
Freedman - L - Schrijver
Many extensions and generalizations
July 2010

14. k-labeled quantum graph:
Computing with graphs
k-labeled quantum graph:
finite formal sum of
k-labeled graphs 1 2
infinite dimensional linear space
July 2010

15. is a commutative algebra with unit element
Computing with graphs
Define products:
is a commutative algebra with unit element
...
July 2010

16. f: graph parameter Computing with graphs Inner product:
extend linearly
July 2010

17. Computing with graphs
Factor out the kernel:
July 2010

18. Computing with graphs
Example 1: - -
f( ) f( ) = 0
July 2010

19. f( ) f( ) f( ) = 0 Computing with graphs if  is an integer Example 2:
- (-1) +
- (-1) +
f( ) f( ) f( ) = 0
July 2010

20. f is reflection positive
Computing with graphs
f is reflection positive
July 2010

21. Computing with graphs Write if for every graph H . Turán: -2 +
Kruskal-Katona: -
Blakley-Roy: -
Sidorenko Conjecture:
(F bipartite)
July 2010

22. - Computing with graphs -4 +2 - + 2 = - + 2 = 2 - + = +  + ≥ 0
- 2  +
- 2
≥ 0
t( ,G) – 2t( ,G) + t( ,G) ≥ 0
Goodman’s Theorem
July 2010

23. Positivstellensatz for graphs?
If a quantum graph x is sum of squares (modulo labels
and isolated nodes), then
Question: Suppose that
Does it follow that
No!
Hatami-Norine
is algorithmically undecidable.
July 2010

24. The main trick in the proof
Kruskal-Katona 1
Goodman
Razborov 2006
Fisher
Bollobás
Lovász-Simonovits
1/2
2/3
3/4 1
Mantel-Turán
July 2010

25. A weak Positivestellensatz
July 2010

26. Which inequalities between densities are valid?
Undecidable, but decidable with an arbitrarily small error.
July 2010

27. Is there always an extremal graph?
Minimize over x0
Real numbers are useful
minimum is not attained
in rationals
Minimize t(C4,G) over graphs
with edge-density 1/2
always >1/16,
arbitrarily close for random graphs
Quasirandom graphs
Graph limits are useful
minimum is not attained
among graphs
July 2010

28. Pixel pictures AG G WG July 2010 0 0 1 0 0 1 1 0 0 0 1 0 0 1 AG G WG
July 2010

29. Limit objects
July 2010

30. Limit objects 1/2 A random graph with 100 nodes and with 2500 edges
July 2010

31. Limit objects
Rearranging the rows and columns
July 2010

32. Limit objects A randomly grown uniform attachment graph with 200 nodes
July 2010

33. Limit objects
(graphons)
July 2010

34. W is essentially unique (up to
Limit objects
For every convergent graph sequence (Gn)
there is a graphon such that LS
Conversely, for every graphon W there is a graph sequence (Gn) such that LS
W is essentially unique (up to
measure-preserving transformation).
BCL
July 2010

35. Semidefinite connection matrices
is positive semidefinite,
f( )=1 and f is multiplicative
is positive semidefinite
July 2010

36. Proof of the weak Positivstellensatz (sketch2)
The optimum of the semidefinite program
minimize
subject to M(f,k) positive semidefinite for all k
f(K1)=1
is 0.
Apply the Duality Theorem of semidefinite programming
July 2010

37. W is essentially unique (up to
Limit objects
For every convergent graph sequence (Gn)
there is a graphon such that LS
Conversely, for every graphon W there is a graph sequence (Gn) such that LS
W is essentially unique (up to
measure-preserving transformation).
BCL
July 2010

38. Is there always an extremal graph?
No, but there is always an extremal graphon.
July 2010