Create Presentation
Download Presentation

Download Presentation

Graph homomorphisms, statistical physics, and quasirandom graphs

Download Presentation
## Graph homomorphisms, statistical physics, and quasirandom graphs

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Graph homomorphisms, statistical physics,**and quasirandom graphs László Lovász Microsoft Research lovasz@microsoft.com Joint work with: Christian Borgs, Jennifer Chayes, Mike Freedman, Jeff Kahn, Lex Schrijver, Vera T. Sós, Balázs Szegedy, Kati Vesztergombi**independent**set coloring triangles Homomomorphism: adjacency-preserving map**probability that random map**is a homomorphism every node in G weighted by 1/|V(G)| Homomorphism density: Homomorphism entropy:**if G has no loops**Examples:**1**1 2 1 1 H 3 3 -1 1/4 1/4 -1 -1 -1 -1 1/4 1/4 -1 3 3 H**atoms are in states (e.g. up or down):**energy of interaction: energy of state: partition function: Hom functions and statistical physics interaction only between neighboring atoms: graph G**sparse G**bounded degree partition function: denseG All weights in H are 1 hard-core model H=Kq, all weights are positive soft-core model**: set of connected graphs**Erdős – Lovász – Spencer Recall:**Kruskal-Katona**Goodman 1 0 1/2 2/3 3/4 1 Bollobás Lovász-Simonovits**small probe**(subgraph) small template (model) large graph**Kruskal-Katona Theorem for triangles:**Turán’s Theorem for triangles: Erdős’s Theorem on quadrilaterals:**Connection matrix (for target graph G):**Connection matrices k-labeled graph: k nodes labeled 1,...,k**...**k=2: ...**reflection**positivity Main Lemma: is positive semidefinite has rank**k=2**Proof of Kruskal-Katona k=1**How much does the positive semidefinite**property capture? ...almost everything!**reflection**positivity equality holds in “generic” case (H has no automorphism) is positive semidefinite has rank Connection matrix of a parameter: graph parameter**...**finite sum is a commutative algebra with unit element Inner product: positive semidefinite suppose = k-labeled quantum graph:**Distance of graphs:**Converse???**Example: Paley graphs**p: prime 1 mod 4 Quasirandom graphs Thomason Chung – Graham – Wilson (Gn: n=1,2,...)is quasirandom, if d(Gn, G(n,p)) 0 a.s. How to see that these graphs are quasirandom?**For a sequence (Gn: n=1,2,...), the following are**equivalent: (Gn) is quasirandom; simple graph F; for F=K2 and C4. Converse if G’ is a random graph. Chung – Graham – Wilson**k=1:**1 p ... ... ... ... p p2 ... ... pk pk+1 Suppose that Want:**k=2:**1 p2 p2 p4 ... ... p2k p2k+2 ... ... ... ...**1**pk ... ... ... ... pk p2k ... ... ... ... ... p|E(G’)| p|E(G)| ... ... ... ... k=deg(v)**density 0.2**0.1n 0.4n 0.2n 0.3n density 0.35 For a sequence (Gn: n=1,2,...), the following are equivalent: d(Gn, G(n,H)) 0; simple graph F; Generalized (quasi)random graphs 0.1 0.5 0.7 0.2 0.3 0.2 0.4 0.5 0.35 0.3**(Gn) left-convergent:**(Gn) right-convergent: Recall:**Example:**(C2n) is right-convergent But... (Cn) is not convergent for bipartite H**(Gn) left-convergent:**e.g. H=Kq, q>8D Graphs with bounded degree D