Graph algebras and extremal graph theory László Lovász April 20131

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3 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)? Vera Sós: How to characterize generalized quasirandom graphs?

April 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)? Vera Sós: How to characterize generalized quasirandom graphs?

April 2013 Graph parameter: isomorphism-invariant function on finite graphs 5 Which parameters are homomorphism functions? Homomorphism: adjacency-preserving map Partition functions of vertex coloring models

April 2013 Probability that random map V(F)  V(G) is a hom 6 Weighted version: Which parameters are homomorphism functions?

April k -labeled graph: k nodes labeled 1,...,k, any number of unlabeled nodes 1 2 Which parameters are homomorphism functions?

April 2013 k=2:... 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  q k 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)

G AGAG WGWG Graphs  Graphons April

April 2013 Limit objects 13 (graphons) t(F,W G )=t(F,G) (G 1,G 2,…) convergent:  simple F t(F,G n ) converges Borgs-Chayes-L-Sós-Vesztergombi

For every convergent graph sequence (G n ) there is a graphon W such that G n  W. For every graphon W there is a graph sequence (G n ) such that G n  W. April 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(K 1 )=1. L-Szegedy

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April 2013 k-labeled quantum graph: finite formal sum of k-labeled graphs 1 2 infinite dimensional linear spaces 16 Computing with graphs G k = {k-labeled quantum graphs} infinite formal sum with  G |x G | L k = {k-labeled quantum graphs (infinite sums)}

April 2013 is a commutative algebra with unit element... Define products: 17 Computing with graphs G 1,G 2 : k-labeled graphs G 1 G 2 = G 1  G 2, labeled nodes identified

April 2013 Inner product: f: graph parameter extend linearly 18 Computing with graphs

April 2013 f is reflection positive Computing with graphs 19

April 2013 Factor out the kernel: 20 Computing with graphs

In particular, for k=1 : Computing with graphs April Assume that rk(M(f,k))=q k.

Computing with graphs April p i  p j form a basis of  2. Assume that rk(M(f,k))=q k.

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April 2013 Turán’s Theorem (special case proved by Mantel): G contains no triangles  #edges  n 2 /4 Theorem (Goodman): t(K 3,G)  (2t(K 2,G) -1) t(K 2,G) Extremal: 24 Some old and new results from extremal graph theory

April 2013 Theorem (Erdős): G contains no 4-cycles  #edges  n 3/2 /2 (Extremal: conjugacy graph of finite projective planes) 25 Some old and new results from extremal graph theory t(C 4,G)  t(K 2,G) 4

April 2013 k=2: 26 Connection matrices t(,G)  t(,G) 2  t(,G) 4

April Which inequalities between densities are valid? Undecidable… Hatami-Norine 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+ Kruskal-Katona: - Blakley-Roy: - Computing with graphs 28

April = = Goodman’s Theorem Computing with graphs  + ≥ = t(,G) – 2t(,G) + t(,G) ≥ 0

April 2013 Question: Suppose that x ≥ 0. Does it follow that 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  A weak Positivstellensatz for graphs 31 L-Szegedy

April 2013 An exact Positivstellensatz for graphs 32  *: graphs partially labeled by different nonnegative integers  : functions f:  *   such that  F |f(F)|< .

April 2013 x  0  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(K 1 )=1 f(G  K 1 )=f(G) April 2013 Proof of the weak Positivstellensatz (sketch 2 ) Apply Duality Theorem of semidefinite programming 34

April Pentagon density in triangle-free graphs

April Conjecture: Erdős 1984 Proof: Hatami-Hladky-Král-Norine-Razborov 2011 Pentagon density in triangle-free graphs

April labeled quantum graphs positive semidefinite Pentagon density in triangle-free graphs

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