1. geometric representations of graphs
Eigenvalues and
geometric representations
of graphs
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA
Cim

2. (all graphs connected)
Matrices associated with graphs
(all graphs connected)
Adjacency matrix:
Laplacian:

3. Adjacency matrix:
Laplacian:

4. Eigenvalues and eigenvectors
eigenvalues of adjacency matrix
eigenvalues of Laplacian

5. Eigenvalues and eigenvectors-1 1 1 -1
1.481
1.193 1

6. The largest eigenvalue
average degree
maximum degree
If G is regular of degree d, then

7. The largest eigenvalue
Wilf
chromatic number
maximum clique

8. The largest eigenvalue
not used!

9. The smallest eigenvalue
G bipartite  1 1 -1

10. The smallest eigenvalue
G k-colorable 
Hoffman
Proof uses only 0 entries:
can replace 1’s by anything
maximizing we get:
Polynomial time
computable!

11. semidefinite optimization problem
Computing
semidefinite optimization problem

12. Another matrix associated with graphs
Adjacency matrix:
Laplacian:
Transition matrix
(of random walk):
(Not much difference if graph is regular.)

13. Random walks
How long does it take to get completely lost?

14. Sampling by random walk
S: large and complicated set
Want: uniformly distributed random element from S
(all lattice points in convex body
all states of a physical system
all matchings in a graph...)
Applications:
- statistics
- simulation
- counting
- numerical integration
- optimization
- card shuffling...

15. One general method for sampling: random walks
(+rejection sampling, lifting,…)
Want: sample from set V
Construct regular connected non-bipartite graph with
node set V
Walk for T steps
????????????
mixing time
Output the final node

16. Example: random linear extension of partial order1 2 5 4 3
Node: compatible
linear order 1 3 5 4 2
Step:
- pick randomly label i<n;
- interchange i and i+1
if possible
Given: poset

17. The second largest eigenvalue

18. Conductance Edge-density in cut frequency of stepping from S to V \ S
in random walk:
in sequence of independent samples:
Edge-density
in cut
conductance:

19. Conductance and eigenvalue gap
eigenvalues of transition matrix
Jerrum - Sinclair
up to a constant factor

20. Conductance and eigenvalue gap
eigenvalues of transition matrix
Jerrum - Sinclair

21. What about the eigenvectors?
eigenvalues of A
G connected  l1 has multiplicity 1
eigenvector is all-positive
Frobenius-Perron

22. What about the eigenvectors?
eigenvalues of A
are connected.
Van der Holst