Transcript Slide 1
Eigenvalues and
geometric representations
of graphs
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA 98052
[email protected]
Matrices associated with graphs
(all graphs connected)
Adjacency matrix:
Laplacian:
A (aij ),
aij # ij edges
L (lij ) ,
aij , if i j ,
lij
di , if i j.
x T Lx ( xi x j ) 2
ij E
Adjacency matrix:
Laplacian:
æ0
çç
çç 1
çç
çç 1
çç
çç 0
çç
è
ç 1
æ
çç
çç çç
çç çç
çç
çç
è
ç -
1 1 0
0 1 0
1 0 1
0 1 0
0 0 1
ö
1÷
÷
÷
0÷
÷
÷
÷
0÷
÷
÷
÷
1÷
÷
÷
÷
0÷
ø
÷
ö
0 - 1÷
÷
÷
1 2 - 1 0 0÷
÷
÷
÷
1 - 1 3 - 1 0÷
÷
÷
÷
0 0 - 1 2 - 1÷
÷
÷
÷
÷
1 0 0 - 1 2ø
÷
3 - 1 - 1
Eigenvalues and eigenvectors
max 1 2 ... n min
eigenvalues of adjacency matrix
1 2 ... n Tr( A) 0
min 0 1 2 ... n max
eigenvalues of Laplacian
Eigenvalues and eigenvectors
5 2
-1
1
0
1
-1
1 2.481...
4 0
1
1.481
0
1
-1
-1
1.193
1
1
1.481
The largest eigenvalue
max max
x T Ax
x
2
daverage max dmax
average degree
dmax max dmax
maximum degree
If G is regular of degree d, then max d .
The largest eigenvalue
(G) max 1
(G) max 1
chromatic number
Note: (G) (G)
Wilf
maximum clique
0
1
1
0
1
0
1
0
1
0
0
1
1
1
0
1
0
0
0
0
1
0
1
0
1
0
0
1
0
1
0
1
0
0
1
0
The largest eigenvalue
0
1
1
0
1
0
1
0
1
0
0
1
1
1
0
1
0
0
0
0
1
0
1
0
1
0
0
1
0
1
0
1
0
0
1
0
(G) max 1
not used!
1
7
0 1 1 .2
1
1 0 1 3 0
1 1 0
1 1.5 0
A ' .2 3 1
0
1 .2
0
1
1 0 1.5 1
0
7 1 0 .2 1
(G) max ( A ') 1
(G) min A' max ( A ') 1
(G) (G)
The smallest eigenvalue
G bipartite
max min
1
1
1
-1
1
1
1
-1
1
1
1
-1
The smallest eigenvalue
G k-colorable
max (k 1)min
max
1
(G) 1 max
min
(G) (G) (G)
Hoffman
Proof uses only 0 entries:
can replace 1’s by anything
maximizing we get: (G)
Polynomial time
computable!
Computing (G)
(G) min A' max ( A ') 1
1
7
0 1 1 .2
1
1 0 1 3 0
1 1 0
1 1.5 0
A ' .2 3 1
0
1 .2
0
1
1 0 1.5 1
0 .2 1
0
7 1
semidefinite
optimization
problem
(G ) minimize t
subject to X ii t 1 (i V )
X ij 1
(ij E )
( X ij ) positive semidefinite
Another matrix associated with graphs
Adjacency matrix:
Laplacian:
Transition matrix
(of random walk):
A (aij ),
aij # ij edges
L (lij ) ,
aij , if i j ,
lij
di , if i j.
P ( pij ),
pij
aij
di
(Not much difference if graph is regular.)
Random walks
How long does it take to get completely lost?
Sampling by random walk
S: large and complicated set
(all lattice random
points inelement
convex from
body S
Want: uniformly distributed
Applications:
- statistics
all states of a physical system
all matchings in a graph...)
- simulation
- counting
- numerical integration
- optimization
- card shuffling...
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
Output the final node
????????????
mixing time
Example: random linear extension of partial order
5
4
5
4
2
3
3
2
1
1
Given: poset
Node: compatible
linear order
Step:
- pick randomly label i<n;
- interchange i and i+1
if possible
The second largest eigenvalue
1
1 1 2 ... n : eigenvalues of P A
d
max{2 , n }
v0 , v1 ,..., vt ,...: random walk
A
Pr(v A)
V
t
V t
A V
Conductance
S
V \S
frequency of stepping from S to V \ S
in random walk:
E (S ,V \ S )
dV
S V \S
in sequence of independent samples:
V
2
V E ( S ,V \ S )
( S )
d
S V \S
Edge-density
conductance: min S (S )
in cut
Conductance and eigenvalue gap
1 1 2 ... n
eigenvalues of transition matrix
2 1 2
Jerrum - Sinclair
up to a constant
factor
Conductance and eigenvalue gap
1 1 2 ... n
eigenvalues of transition matrix
2 1 2
Jerrum - Sinclair
T
x
Ax T
2 max 2 : 1 x 0
x
T
T
x Px T
1
x Lx T
1 2 min 1
: 1 x 0 min 2 : 1 x 0
2
x
d
x
x T Lx ( xi x j ) 2
ij E
What about the eigenvectors?
1 2 ... n
eigenvalues of A
G connected l1 has multiplicity 1
eigenvector is all-positive
Frobenius-Perron
What about the eigenvectors?
1 2 ... n
eigenvalues of A
Ax 2 x
x0
x0
supp ( x), supp ( x) are connected.
x0
Van der Holst