Extremal graph theory and limits of graphs Lszl
- Slides: 58
Extremal graph theory and limits of graphs László Lovász September 2012 1
Some old and new results from extremal graph theory Turán’s Theorem (special case proved by Mantel): G contains no triangles #edges n 2/4 Extremal: Theorem (Goodman): September 2012 2
Some old and new results from extremal graph theory Kruskal-Katona Theorem (very special case): k September 2012 n 3
Semidefiniteness and new extremal graph theory Some old and results from Trickytheory examples extremal graph Kruskal-Katona 1 Razborov 2006 Fisher Goodman 0 1/2 2/3 3/4 Bollobás Mantel-Turán September 2012 1 Lovász-Simonovits 4
Some old and new results from extremal graph theory Theorem (Erdős): G contains no 4 -cycles #edges n 3/2/2 (Extremal: conjugacy graph of finite projective planes) September 2012 5
Some old and new results from extremal graph theory Theorem (Erdős-Stone-Simonovits): (F)=3 September 2012 6
General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Is there always an extremal graph? - Which graphs are extremal? September 2012 7
General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Is there always an extremal graph? - Which graphs are extremal? September 2012 8
Homomorphism functions Homomorphism: adjacency-preserving map Probability that random map V(F) V(G) is a hom If valid for large G, then valid for all September 2012 9
General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Is there always an extremal graph? - Which graphs are extremal? September 2012 10
Which inequalities between densities are valid? Undecidable… Hatami-Norine September 2012 11
The main trick in the proof 1 0 t( , G) – 2 t( September 2012 1/2 2/3 3/4 , G) + t( , G) = 0 1 … 12
Which inequalities between densities are valid? Undecidable… Hatami-Norine …but decidable with an arbitrarily small error. L-Szegedy September 2012 13
General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Is there always an extremal graph? - Which graphs are extremal? September 2012 14
Which parameters are homomorphism functions? Graph parameter: isomorphism-invariant function on finite graphs k-labeled graph: k nodes labeled 1, . . . , k, any number of unlabeled nodes 1 2 September 2012 15
Connection matrices M(f, k) k=2: . . . September 2012 16
Which parameters are homomorphism functions? f = hom(. , H) for some weighted graph H M(f, k) is positive semidefinite and has rank ck Freedman - L - Schrijver September 2012 17
Computing with graphs k-labeled quantum graph: 1 finite formal sum of k-labeled graphs 2 Gk = {k-labeled quantum graphs} infinite dimensional linear space September 2012 18
Computing with graphs Define products: G 1, G 2: k-labeled graphs G 1 G 2 = G 1 G 2, labeled nodes identified is a commutative algebra with unit element September 2012 . . . 19
Computing with graphs Inner product: f: graph parameter extend linearly September 2012 20
Computing with graphs f is reflection positive September 2012 21
Computing with graphs Write x ≥ 0 if hom(x, G) ≥ 0 for every graph G. Turán: -2 + Kruskal-Katona: Blakley-Roy: September 2012 - - 22
Computing with graphs 2 - - + = 2 - - 2 +2 + - 2 = -2 t( + , G) – 2 t( September 2012 ≥ 0 , G) + t( , G) ≥ 0 - - =2 +2 -4 - - -2 + + +2 -4 +2 + Goodman’s Theorem 23
Positivstellensatz for graphs? If a quantum graph x is sum of squares (ignoring labels and isolated nodes), then x ≥ 0. Question: Suppose that x ≥ 0. Does it follow that No! Hatami-Norine September 2012 24
A weak Positivstellensatz Let x be a quantum graph. Then x 0 L-Szegedy September 2012 25
Proof of the weak Positivstellensatz (sketch 2) 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) Apply Duality Theorem of semidefinite programming September 2012 26
General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Is there always an extremal graph? - Which graphs are extremal? September 2012 27
Is there always an extremal graph? Minimize over x 0 minimum is not attained in rationals Real numbers are useful Minimize t(C 4, G) over graphs with edge-density 1/2 s h p a r g alwaysm>1/16, o d n arbitrarily close a for random graphs r i as u Q Graph limits minimum is not attained are useful among graphs September 2012 28
Limit objects (graphons) September 2012 29
Graphs Graphons G 0 0 1 1 0 0 0 1 0 1 0 0 0 1 0 1 1 1 1 0 0 1 0 1 0 1 1 0 0 0 1 1 0 1 0 1 1 1 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 1 1 0 0 0 0 1 1 0 1 0 1 0 1 1 1 0 0 0 1 1 0 1 0 0 1 1 0 0 0 1 1 1 0 1 0 0 1 0 AG WG September 2012 30
Limit objects (graphons) t(F, WG)=t(F, G) (G 1, G 2, …) convergent: F t(F, Gn) converges September 2012 31
Limit objects For every convergent graph sequence (Gn) there is a graphon W such that Gn W. LS For every graphon W there is a graph sequence (Gn) such that Gn W. LS W is essentially unique (up to measure-preserving transformation). BCL September 2012 32
Is there always an extremal graph? No, but there is always an extremal graphon. The space of graphons is compact. September 2012 33
Semidefinite connection matrices f: graph parameter f = t(. , W) k M(f, k) is positive semidefinite, f( )=1 and f is multiplicative September 2012 34
General questions about extremal graphs -Which inequalities between subgraph densities are valid? - Can all valid inequalities be proved using just Cauchy-Schwarz? - Is there always an extremal graph? - Which graphs are extremal? September 2012 35
Extremal graphon problem Given quantum graphs g 0, g 1, …, gm, find max t(g 0, W) subject to t(g 1, W) = 0 … t(gm, W) = 0 September 2012 36
Finite forcing Finitely forcible graphons Graphon W is finitely forcible: Every finitely forcible graphon is extremal: minimize Every unique extremal graphon is finitely forcible. ? ? Every extremal graph problem has a finitely forcible extremal graphon ? ? September 2012 37
Finitely forcible graphons Goodman 1/2 September 2012 Graham. Chung. Wilson 38
Which graphs are extremal? Stepfunction: Stepfunctions finite graphs with node and edgeweights Stepfunctions are finitely forcible September 2012 L – V. T. Sós 39
Finitely expressible properties d-regular graphon: d-regular September 2012 40
Finitely expressible properties W is 0 -1 valued, and can be rearranged to be monotone decreasing in both variables "W is 0 -1 valued" is not finitely expressible in terms of simple gaphs. W is 0 -1 valued September 2012 41
Finitely forcible graphons ? p(x, y)=0 p monotone decreasing symmetric polynomial finitely forcible September 2012 42
Finitely forcible graphons S p(x, y)=0 Stokes September 2012 43
Finitely forcible graphons Is the following graphon finitely forcible? angle <π/2 September 2012 44
The Simonovits-Sidorenko Conjecture F bipartite, G arbitrary t(F, G) ≥ t(K 2, G)|E(F)| Known when September 2012 ? F is a tree, cycle, complete bipartite… Sidorenko F is hypercube Hatami F has a node connected to all nodes in the other color class Conlon, Fox, Sudakov F is "composable" Li, Szegedy 45
The Simonovits-Sidorenko Conjecture Two extremal problems in one: For fixed G and |E(F)|, t(F, G) is minimized by F= … For fixed F and t( , G), t(F, G) is minimized by random G September 2012 asymptotically 46
The integral version Let W W 0, W≥ 0, ∫W=1. Let F be bipartite. Then t(F, W)≥ 1. ? For fixed F, t(F, W) is minimized over W≥ 0, ∫W=1 by W 1 September 2012 47
The local version Let Then t(F, W) 1. September 2012 48
The idea of the proof 0 0< September 2012 49
The idea of the proof Main Lemma: If -1≤ U ≤ 1, shortest cycle in F is C 2 r, then t(F, U) ≤ t(C 2 r, U). September 2012 50
Common graphs Erdős: ? Thomason September 2012 51
Common graphs Hatami, Hladky, Kral, Norine, Razborov F common: Common graphs: Non-common graphs: Sidorenko graphs (bipartite? ) graph containing September 2012 Jagger, Stovícek, Thomason 52
Common graphs September 2012 53
Common graphs F common: is common. 8 +2 + =4 +2 September 2012 Franek-Rödl +4 +( +2 )2 +4( - ) 54
Common graphs F locally common: is locally common. 12 +3 12 2 September 2012 +3 +3 2 +12 +3 4 Franek-Rödl + +12 4 + 6 55
Common graphs graph containing is locally common but not common. Not locally common: September 2012 56
Common graphs F common: is common. 8 - 1/2 +2 + +4 September 2012 Franek-Rödl =4 1/2 +2 +( -2 )2 57
Common graphs Hatami, Hladky, Kral, Norine, Razborov F common: Common graphs: Non-common graphs: Sidorenko graphs (bipartite? ) graph containing September 2012 Jagger, Stovícek, Thomason 58
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