1. Graph homomorphisms, statistical physics,
and quasirandom graphs
László Lovász
Microsoft Research
Joint work with:
Christian Borgs, Jennifer Chayes, Mike Freedman,
Jeff Kahn, Lex Schrijver, Vera T. Sós,
Balázs Szegedy, Kati Vesztergombi

2. Homomomorphism: adjacency-preserving map
coloring
independent
set
triangles

3. Weighted version:

4. G connected 

5. L1966

6. probability that random map
Homomorphism density:
probability that random map
is a homomorphism
every node in G
weighted by 1/|V(G)|
Homomorphism entropy:

7. Examples:
if G has no loops

8. 3 3 -1
1/4
1/4 -1 -1 -1 -1
1/4
1/4 -1 3 3 H 1 2 H

9. Hom functions and statistical physics
atoms are in states (e.g. up or down):
interaction only between neighboring atoms: graph G
energy of interaction:
energy of state:
partition function:

10. H=Kq, all weights are positive  soft-core model
sparse G
bounded degree
partition function:
All weights in H are 1  hard-core model
H=Kq, all weights are positive  soft-core model
dense G

11. Recall:
: set of connected graphs
Erdős – Lovász – Spencer

12. Kruskal-Katona 1
Goodman
1/2
2/3
3/4 1
Bollobás
Lovász-Simonovits

13. small probe
(subgraph)
small template
(model)
large graph

14. Turán’s Theorem for triangles:
Kruskal-Katona Theorem for triangles:
Erdős’s Theorem on quadrilaterals:

15. k-labeled graph: k nodes labeled 1,...,k
Connection matrices
k-labeled graph: k nodes labeled 1,...,k
Connection matrix (for target graph G):

16. ...
k=2:
...

17. Main Lemma:
is positive semidefinite
reflection
positivity
has rank

18. Proof of Kruskal-Katona

19. How much does the positive semidefinite
property capture?
...almost everything!

20. Connection matrix of a parameter:
graph parameter
is positive semidefinite
has rank
reflection
positivity
equality holds in “generic” case
(H has no automorphism)

21. k-labeled quantum graph:
finite sum
is a commutative algebra with unit element
...
Inner product:
positive
semidefinite
suppose =

23. Distance of graphs:
Converse???

24. (Gn: n=1,2,...) is quasirandom, if d(Gn, G(n,p))  0 a.s.
Quasirandom graphs
Thomason
Chung – Graham – Wilson
(Gn: n=1,2,...) is quasirandom, if d(Gn, G(n,p))  0 a.s.
Example: Paley graphs
p: prime 1 mod 4
How to see that these graphs are quasirandom?

25. For a sequence (Gn: n=1,2,...), the following are equivalent:
(Gn) is quasirandom;
 simple graph F;
for F=K2 and C4.
Chung – Graham – Wilson
Converse if G’ is a random graph.

27. Suppose that
Want:
k=1:
...
... 1 p p2 pk
pk+1

28. k=2:
...
... 1 p2 p4
p2k
p2k+2

29. k=deg(v) 1 pk p2k p|E(G’)| p|E(G)| ... ... ... ... ... ... ... ... ...

30. Generalized (quasi)random graphs
0.1
0.1n
0.2n
0.3n
0.4n
density 0.2
0.5
0.7
0.2
0.3
0.2
0.4
0.5
0.35
0.3
density 0.35
For a sequence (Gn: n=1,2,...), the following are equivalent:
d(Gn, G(n,H))  0;
 simple graph F;

31. (Gn) left-convergent:
Recall:
(Gn) left-convergent:
(Gn) right-convergent:

32. (C2n) is right-convergent
Example:
(C2n) is right-convergent
But... (Cn) is not convergent
for bipartite H

33. Any connection between left and right convergence?

34. (Gn) left-convergent:
Graphs with bounded degree D
(Gn) left-convergent:
e.g. H=Kq, q>8D