Download CS 598: Spectral Graph Theory: Lecture 3

CS 598: Spectral Graph
Theory. Lecture 3
Extremal Eigenvalues and
Eigenvectors of the Laplacian and
the Adjacency Matrix.
Today
More on Courant-Fischer and Rayleigh
quotients
 Applications of Courant-Fischer
 Adjacency matrix vs. Laplacian
 Perron-Frobenius


Courant-Fischer Refresher (1)
xT Ax
 k  max
min T
n
S  R ,dim( S )  k xS
x x
xT Ax
 k  n min
max T
S  R ,dim( S )  n  k 1 xS
x x

Sylvester’s Law of Inertia
xT Ax
 k  max
min T
n
S  R ,dim( S )  k xS
x x

Courant-Fischer Refresher (2)
xT Lx
k  min max T
S of dimk xS
x x
xT Lx
k  max min T
S of dimn  k 1 xS x x
Courant-Fischer for Laplacian
Applying Courant-Fischer for the Laplacian
we get :
  0, v  1

1
2  min
x 1, x  0
1
T
 (x
i
xT Lx
k  min max T
S of dim k xS
x x
 x j )2
x Lx
( i , j )E

min
x 1, x  0
xT x
 xi 2
iV
max  max
x0
T
x Lx
 max
x0
xT x
 (x
( i , j )E


 x j )2
x
iV

i
2
i
Useful for getting bounds, if calculating spectra is cumbersome.
To get upper bound on λ2, just need to produce vector with small
Rayleigh Quotient.
Similarly, t o get lower bound on λmax, just need to produce vector
with large Rayleigh Quotient
Example 1

Lemma1: Let G=(V,E) be a graph with some
vertex w having degree d. Then
max  d

Lemma 2: We can also improve on that. Under
same assumptions, we can show:
max  d  1
Proof: see blackboard
Example 1

Lemma1: Let G=(V,E) be a graph with some
vertex w having degree d. Then
max  d

Lemma 2: We can also improve on that. Under
same assumptions, we can show:
max  d  1
Lemma 2 is tight, take star graph (ex)
Example 2

The Path graph Pn on n vertices has
12
2  2
n

Already knew that, but this is easier and more
general.
Proof: see blackboard
Example 3

2
2 
n
Example 3
node 1
node 2
node 4
node 5
node 3
node 1
node 6
0
-1
1
1
1
-1
1
1
1
1
-1
-1
-1
-1
-1

Lower bounds are harder, we will see
some in two lectures (different
technique)
Adjacency Matrix vs.
Laplacian
Adjacency Matrix Refresher
i
j
1, if (i, j ) edge

Aij  
0 if no edge between i, j
j
G = {V,E}
• Unweighted graphs for simplicity
i
A has n eigenvalues (counting multiplicities)
{a1 a2  … an }
1
Adjacency Matrix vs. Laplacian for
d-regular graphs

Bounds on the Eigenvalues of
Adjacency Matrix

Bounds on the Eigenvalues of
Adjacency Matrix

Courant-Fischer for Adjacency
Matrix Refresher
xT Ax
 k  max min T
S of dim k xS
x x
xT Ax
1  max T
x 0
x x

Will see next how to apply Courant-Fischer for the adjacency
matrix to get another bound on the first eigenvalue as well as a
relation to graph coloring
Bounding Adjacency Matrix
Eigenvalues

xT Ax
1  max T
x 0
x x
Bounding Adjacency Matrix
Eigenvalues

Chromatic Number

Chromatic Number


Adjacency Matrix: The PerronFrobenius Theorem
Adjacency Matrix: The PerronFrobenius Theorem

Proof in 3 parts.
We next show
Lemma 1: If all entries of an nxn
matrix A are positive, then it has a positive
eigenvector v with corresponding positive
eigenvalue α, that is Av=αv.

Geometric proof on blackboard

Adjacency Matrix: The PerronFrobenius Theorem
Adjacency Matrix: The PerronFrobenius Theorem

Laplacian: The Perron-Frobenius
Theorem

Theory can also be applied to Laplacians and any
matrix with non-positive off-diagonal entries. It
involves the eigenvector with smallest eigenvalue.
Perron-Frobenius for Laplacians:Let M be a matrix
with non-positive off-diagonal entries s.t. the graph
of the no-zero off-diagonal entries is connected.
Then the smallest eigenvalue has multiplicity 1 and
the corresponding eigenvector is strictly positive.
Proof on blackboard.