Lecture 3: Eigenvalues of the Laplacian
Transcriber: Andy Parrish
In this lecture we will consider only graphs G = (V, E) with no isolated vertices and no selfloops.
Recall that
A
is the adjacency matrix of a graph, and
D
is the diagonal matrix of degrees.
Let
M
=
D

1
/
2
AD

1
/
2
.
Define the Laplacian of a graph to be
L
=
I

M
=
D

1
/
2
(
D

A
)
D

1
/
2
. Compare this with the
combinatorial Laplacian
L
=
D

A
. We see
L
=
D

1
/
2
LD

1
/
2
.
We see
L
has entries
L
(
u, v
) =
1
if
u
=
v

1
√
d
u
d
v
if
u
∼
v
0
otherwise
Since
L
is symmetric, we can call its eigenvalues
λ
0
≤
λ
1
≤
. . .
≤
λ
n

1
.
We may compute the
eigenvalues by Rayleigh quotients:
R
(
g
) =
< g,
L
g >
< g, g >
=
gD

1
/
2
LD

1
/
2
g
*
gg
*
Since we may write any vector
g
as
fD
1
/
2
, we get
R
(
fD
1
/
2
) =
fLf
*
fDf
*
Now, we compute
fLf
*
=
X
v
f
(
v
)
2
d
v

X
u,v
f
(
u
)
A
(
u, v
)
!
f
(
v
)
=
X
v
X
u
∼
v
(
f
(
v
)

f
(
u
))
f
(
v
)
=
X
uv
∈
E
(
f
(
v
)

f
(
u
))
2
where this last equality comes because we are counting each edge twice: once at
v
and once at
u
Alternately, we can get the same answer using the matrix
B
from the proof of the Matrix Tree
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 Fall '08
 aterras
 Math, Linear Algebra, Matrices, Eigenvalue, eigenvector and eigenspace, Singular value decomposition, Orthogonal matrix, uv ∈E

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