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:
ж
зз
зз
зз
зз
зз
зз
зз
из
0 1 1 0 1 цч
чч
1 0 1 0 0 чч
чч
1 1 0 1 0 чч
чч
0 0 1 0 1 чч
чч
чч
1 0 0 1 0ш
ж 3 - 1 - 1 0 - 1 цч
зз
ч
зз - 1 2 - 1 0 0 чч
чч
зз
зз - 1 - 1 3 - 1 0 ччч
чч
зз
зз 0 0 - 1 2 - 1 чч
чч
зз
чч
из - 1 0 0 - 1 2 ш
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
-1
1
0
1
-1
1  2.481...
4  0
 5  2
1
1.481
0
1
-1
-1
1.193
1
1
1.481
The largest eigenvalue
max  max
x T Ax
x
2
d average   max  d max
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
.2
1
7 
0 1
1 3 0
1 
1 0
0
1 1.5 0 
A'   1 1
0
1 .2
 .2 3 1
0
1 
 1 0 1.5 1
0 .2 1
0 
7 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
.2
1
7 
0 1
1 3 0
1 
1 0
1
1
0
1
1.5
0


A' 
0
1 .2
 .2 3 1
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
x0
x0
supp ( x), supp ( x) are connected.
x0
Van der Holst