1. Graph Algebras László Lovász Eötvös University, Budapest
Joint work with:
Christian Borgs, Jennifer Chayes, Mike Freedman,
Jeff Kahn, Lex Schrijver, Vera T. Sós,
Balázs Szegedy, Kati Vesztergombi, Dominic Welsh

2. k-labeled graph: k nodes labeled 1,...,k,2
k-labeled graph: k nodes labeled 1,...,k,
any number of unlabeled nodes
k-labeled quantum graph:
finite sum of
k-labeled graphs 1 2
infinite dimensional linear space

3. is a commutative algebra with unit element
Define products:
is a commutative algebra with unit element
...

4. Inner product: f: graph parameter extend linearly
f is reflection positive:

5. Factor out the kernel:

6. Example 1: - -
f( ) f( ) = 0

7. Example 2:
if  is an integer
- (-1) +
- (-1) +
f( ) f( ) f( ) = 0

8. Connection matrices M(f, k)
...
k=2:
...

9. For which parameters f is finite?
For which parameters f is semidefinite?

10. Observation:
If is finite, then f(G) can be evaluated in
polynomial time for graphs with tree-width at most k.
L- Welsh

11. Homomorphism: adjacency-preserving map
coloring
independent
set
triangles

12. Probability that random map
V(G)V(H) is a hom
Weighted version:

13. Examples:
hom(G, ) = # of independent sets in G
if G has no loops

14. H H partition functions in statistical physics... 3 3 -1 1/4 1/4 -1 -12 H
partition functions in statistical physics...

15. M(f,0) has rank 1 and M(f,2) has finite rank. is positive semidefinite
and has rank
Freedman - L - Schrijver
Enough to assume that
M(f,0) has rank 1 and
M(f,2) has finite rank.
L-Szegedy
Difficult direction:
Easy but useful direction:

17. What is the dimension of ?
If H has no "twins":

19. Computations in the algebra of graphs- + 2 = - + - + 2 +2 = - + +2 -4
Turán's Theorem
for triangles

20. For write if for every weighted
graph H .
Turán: -2 +
Kruskal-Katona: -
Blakley-Roy: -
Sidorenko Conjecture:
(F bipartite)

21. Question: Suppose that .
Does it follow that
Positivstellensatz
for graphs?

22. Almost...
graph parameter
reflection
positive
L - B. Szegedy
[without (a)? finite?]

23. Edge coloring models
number of perfect matchings, number of edge-colorings,...?
Given
For

24. number of
perfect matchings
number of
3-edge-colorings

25. k-broken quantum graph:1 2
k-broken graph: k half-edges any number of
full edges and nodes 1 2
k-broken quantum graph: 1 2
finite sum of
k-broken graphs

26. Product: =

27. Inner product:
f: graph parameter
extend linearly

28. B. Szegedy
Where is the finite dimension condition?
It follows!
Where is the number of colors?

29. What is the dimension of Bk/h?
tensor

30. The dimension of Bk is the dimension all tensors
invariant under Ort(H).
Schrijver