Eigenvalues and

1
Eigenvalues and geometric representations of
graphs László Lovász Microsoft Research One
Microsoft Way, Redmond, WA 98052
lovasz_at_microsoft.com
2
Matrices associated with graphs
(all graphs connected)
3
(No Transcript)
4
Eigenvalues and eigenvectors
5
Eigenvalues and eigenvectors
6
The largest eigenvalue
7
The largest eigenvalue
chromatic number
maximum clique
8
The largest eigenvalue
9
The smallest eigenvalue
10
The smallest eigenvalue
maximizing we get
Polynomial time computable!
11
Computing
semidefinite optimization problem
12
Another matrix associated with graphs
Adjacency matrix
Laplacian
(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
(all lattice points in convex body all states of
a physical system all matchings in a graph...)
Want uniformly distributed random element from S
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 order
17
The second largest eigenvalue
18
Conductance
frequency of stepping from S to V \ S
Edge-density in cut
19
Conductance and eigenvalue gap
eigenvalues of transition matrix
up to a constant factor
20
Conductance and eigenvalue gap
eigenvalues of transition matrix
21
What about the eigenvectors?
eigenvalues of A
22
What about the eigenvectors?
eigenvalues of A