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

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

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

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

Factor out the kernel:

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

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

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

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

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

Homomorphism: adjacency-preserving map coloring independent set triangles

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

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

H H partition functions in statistical physics... 3 3 -1 1/4 1/4 -1 -1 2 H partition functions in statistical physics...

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:

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

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

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

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

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

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

number of perfect matchings number of 3-edge-colorings

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

Product: =

Inner product: f: graph parameter extend linearly

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

What is the dimension of Bk/h? tensor

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