<s>
In	O
the	O
mathematical	O
field	O
of	O
graph	O
theory	O
,	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
,	O
also	O
called	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
,	O
admittance	O
matrix	B-Architecture
,	O
Kirchhoff	B-Algorithm
matrix	I-Algorithm
or	O
discrete	B-Algorithm
Laplacian	I-Algorithm
,	O
is	O
a	O
matrix	B-Architecture
representation	O
of	O
a	O
graph	O
.	O
</s>
<s>
Named	O
after	O
Pierre-Simon	O
Laplace	O
,	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
matrix	B-Architecture
can	O
be	O
viewed	O
as	O
a	O
matrix	B-Architecture
form	O
of	O
the	O
negative	O
discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
on	O
a	O
graph	O
approximating	O
the	O
negative	O
continuous	O
Laplacian	O
obtained	O
by	O
the	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
The	O
Laplacian	B-Algorithm
matrix	I-Algorithm
relates	O
to	O
many	O
useful	O
properties	O
of	O
a	O
graph	O
.	O
</s>
<s>
The	O
sparsest	O
cut	O
of	O
a	O
graph	O
can	O
be	O
approximated	O
through	O
the	O
Fiedler	O
vector	O
—	O
the	O
eigenvector	O
corresponding	O
to	O
the	O
second	O
smallest	O
eigenvalue	O
of	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
—	O
as	O
established	O
by	O
Cheeger	O
's	O
inequality	O
.	O
</s>
<s>
The	O
spectral	O
decomposition	O
of	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
allows	O
constructing	O
low	O
dimensional	O
embeddings	O
that	O
appear	O
in	O
many	O
machine	O
learning	O
applications	O
and	O
determines	O
a	O
spectral	O
layout	O
in	O
graph	O
drawing	O
.	O
</s>
<s>
Graph-based	O
signal	O
processing	O
is	O
based	O
on	O
the	O
graph	O
Fourier	O
transform	O
that	O
extends	O
the	O
traditional	O
discrete	B-Algorithm
Fourier	I-Algorithm
transform	I-Algorithm
by	O
substituting	O
the	O
standard	O
basis	O
of	O
complex	O
sinusoids	O
for	O
eigenvectors	O
of	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
of	I-Algorithm
a	I-Algorithm
graph	I-Algorithm
corresponding	O
to	O
the	O
signal	O
.	O
</s>
<s>
The	O
Laplacian	B-Algorithm
matrix	I-Algorithm
is	O
the	O
easiest	O
to	O
define	O
for	O
a	O
simple	O
graph	O
,	O
but	O
more	O
common	O
in	O
applications	O
for	O
an	O
edge-weighted	O
graph	O
,	O
i.e.	O
,	O
with	O
weights	O
on	O
its	O
edges	O
—	O
the	O
entries	O
of	O
the	O
graph	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Spectral	O
graph	O
theory	O
relates	O
properties	O
of	O
a	O
graph	O
to	O
a	O
spectrum	O
,	O
i.e.	O
,	O
eigenvalues	O
,	O
and	O
eigenvectors	O
of	O
matrices	O
associated	O
with	O
the	O
graph	O
,	O
such	O
as	O
its	O
adjacency	B-Algorithm
matrix	I-Algorithm
or	O
Laplacian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Imbalanced	O
weights	O
may	O
undesirably	O
affect	O
the	O
matrix	B-Architecture
spectrum	O
,	O
leading	O
to	O
the	O
need	O
of	O
normalization	O
—	O
a	O
column/row	O
scaling	O
of	O
the	O
matrix	B-Architecture
entries	O
—	O
resulting	O
in	O
normalized	O
adjacency	O
and	O
Laplacian	O
matrices	O
.	O
</s>
<s>
where	O
D	O
is	O
the	O
degree	B-Algorithm
matrix	I-Algorithm
and	O
A	O
is	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
graph	O
.	O
</s>
<s>
Here	O
is	O
a	O
simple	O
example	O
of	O
a	O
labelled	O
,	O
undirected	O
graph	O
and	O
its	O
Laplacian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
We	O
observe	O
for	O
the	O
undirected	O
graph	O
that	O
both	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
and	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
are	O
symmetric	B-Algorithm
,	O
and	O
that	O
row	O
-	O
and	O
column-sums	O
of	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
are	O
all	O
zeros	O
.	O
</s>
<s>
In	O
the	O
directed	O
graph	O
,	O
both	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
and	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
are	O
asymmetric	O
.	O
</s>
<s>
In	O
its	O
Laplacian	B-Algorithm
matrix	I-Algorithm
,	O
column-sums	O
or	O
row-sums	O
are	O
zero	O
,	O
depending	O
on	O
whether	O
the	O
indegree	O
or	O
outdegree	O
has	O
been	O
used	O
.	O
</s>
<s>
where	O
is	O
the	O
matrix	B-Architecture
transpose	O
of	O
B	O
.	O
</s>
<s>
An	O
alternative	O
product	O
defines	O
the	O
so-called	O
edge-based	O
Laplacian	O
,	O
as	O
opposed	O
to	O
the	O
original	O
commonly	O
used	O
vertex-based	O
Laplacian	B-Algorithm
matrix	I-Algorithm
L	O
.	O
</s>
<s>
The	O
Laplacian	B-Algorithm
matrix	I-Algorithm
of	O
a	O
directed	O
graph	O
is	O
by	O
definition	O
generally	O
non-symmetric	O
,	O
while	O
,	O
e.g.	O
,	O
traditional	O
spectral	B-Algorithm
clustering	I-Algorithm
is	O
primarily	O
developed	O
for	O
undirected	O
graphs	O
with	O
symmetric	B-Algorithm
adjacency	O
and	O
Laplacian	O
matrices	O
.	O
</s>
<s>
A	O
trivial	O
approach	O
to	O
apply	O
techniques	O
requiring	O
the	O
symmetry	O
is	O
to	O
turn	O
the	O
original	O
directed	O
graph	O
into	O
an	O
undirected	O
graph	O
and	O
build	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
for	O
the	O
latter	O
.	O
</s>
<s>
In	O
the	O
matrix	B-Architecture
notation	O
,	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
undirected	O
graph	O
could	O
,	O
e.g.	O
,	O
be	O
defined	O
as	O
a	O
Boolean	O
sum	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
original	O
directed	O
graph	O
and	O
its	O
matrix	B-Architecture
transpose	O
,	O
where	O
the	O
zero	O
and	O
one	O
entries	O
of	O
are	O
treated	O
as	O
logical	O
,	O
rather	O
than	O
numerical	O
,	O
values	O
,	O
as	O
in	O
the	O
following	O
example	O
:	O
</s>
<s>
A	O
vertex	O
with	O
a	O
large	O
degree	O
,	O
also	O
called	O
a	O
heavy	O
node	O
,	O
results	O
in	O
a	O
large	O
diagonal	O
entry	O
in	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
dominating	O
the	O
matrix	B-Architecture
properties	O
.	O
</s>
<s>
Normalization	O
is	O
aimed	O
to	O
make	O
the	O
influence	O
of	O
such	O
vertices	O
more	O
equal	O
to	O
that	O
of	O
other	O
vertices	O
,	O
by	O
dividing	O
the	O
entries	O
of	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
by	O
the	O
vertex	O
degrees	O
.	O
</s>
<s>
The	O
symmetrically	O
normalized	O
Laplacian	B-Algorithm
matrix	I-Algorithm
is	O
defined	O
as	O
:	O
</s>
<s>
The	O
symmetrically	O
normalized	O
Laplacian	B-Algorithm
matrix	I-Algorithm
is	O
symmetric	B-Algorithm
if	O
and	O
only	O
if	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
For	O
a	O
non-symmetric	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
a	O
directed	O
graph	O
,	O
either	O
of	O
indegree	O
and	O
outdegree	O
can	O
be	O
used	O
for	O
normalization	O
:	O
</s>
<s>
The	O
left	O
(	O
random-walk	O
)	O
normalized	O
Laplacian	B-Algorithm
matrix	I-Algorithm
is	O
defined	O
as	O
:	O
</s>
<s>
The	O
left	O
or	O
right	O
normalized	O
Laplacian	B-Algorithm
matrix	I-Algorithm
is	O
not	O
symmetric	B-Algorithm
if	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
is	O
symmetric	B-Algorithm
,	O
except	O
for	O
the	O
trivial	O
case	O
of	O
all	O
isolated	O
vertices	O
.	O
</s>
<s>
The	O
example	O
also	O
demonstrates	O
that	O
if	O
has	O
no	O
isolated	O
vertices	O
,	O
then	O
right	B-Algorithm
stochastic	I-Algorithm
and	O
hence	O
is	O
the	O
matrix	B-Architecture
of	O
a	O
random	O
walk	O
,	O
so	O
that	O
the	O
left	O
normalized	O
Laplacian	O
has	O
each	O
row	O
summing	O
to	O
zero	O
.	O
</s>
<s>
In	O
the	O
less	O
uncommonly	O
used	O
right	O
normalized	O
Laplacian	O
each	O
column	O
sums	O
to	O
zero	O
since	O
is	O
left	B-Algorithm
stochastic	I-Algorithm
.	O
</s>
<s>
For	O
a	O
non-symmetric	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
a	O
directed	O
graph	O
,	O
one	O
also	O
needs	O
to	O
choose	O
indegree	O
or	O
outdegree	O
for	O
normalization	O
:	O
</s>
<s>
The	O
left	O
out-degree	O
normalized	O
Laplacian	O
with	O
row-sums	O
all	O
0	O
relates	O
to	O
right	B-Algorithm
stochastic	I-Algorithm
,	O
while	O
the	O
right	O
in-degree	O
normalized	O
Laplacian	O
with	O
column-sums	O
all	O
0	O
contains	O
left	B-Algorithm
stochastic	I-Algorithm
.	O
</s>
<s>
In	O
spectral	B-Algorithm
clustering	I-Algorithm
and	O
graph-based	O
signal	O
processing	O
,	O
where	O
graph	O
vertices	O
represent	O
data	O
points	O
,	O
the	O
edge	O
weights	O
can	O
be	O
computed	O
,	O
e.g.	O
,	O
as	O
inversely	O
proportional	O
to	O
the	O
distances	O
between	O
pairs	O
of	O
data	O
points	O
,	O
leading	O
to	O
all	O
weights	O
being	O
non-negative	O
with	O
larger	O
values	O
informally	O
corresponding	O
to	O
more	O
similar	O
pairs	O
of	O
data	O
points	O
.	O
</s>
<s>
where	O
D	O
is	O
the	O
degree	B-Algorithm
matrix	I-Algorithm
and	O
A	O
is	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
graph	O
.	O
</s>
<s>
Graph	O
self-loops	O
,	O
manifesting	O
themselves	O
by	O
non-zero	O
entries	O
on	O
the	O
main	O
diagonal	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
are	O
allowed	O
but	O
do	O
not	O
affect	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
values	O
.	O
</s>
<s>
For	O
graphs	O
with	O
weighted	O
edges	O
one	O
can	O
define	O
a	O
weighted	O
incidence	B-Algorithm
matrix	I-Algorithm
B	O
and	O
use	O
it	O
to	O
construct	O
the	O
corresponding	O
symmetric	B-Algorithm
Laplacian	O
as	O
.	O
</s>
<s>
An	O
alternative	O
cleaner	O
approach	O
,	O
described	O
here	O
,	O
is	O
to	O
separate	O
the	O
weights	O
from	O
the	O
connectivity	O
:	O
continue	O
using	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
as	O
for	O
regular	O
graphs	O
and	O
introduce	O
a	O
matrix	B-Architecture
just	O
holding	O
the	O
values	O
of	O
the	O
weights	O
.	O
</s>
<s>
We	O
now	O
also	O
define	O
a	O
diagonal	O
matrix	B-Architecture
W	O
containing	O
the	O
edge	O
weights	O
.	O
</s>
<s>
where	O
is	O
the	O
matrix	B-Architecture
transpose	O
of	O
B	O
.	O
</s>
<s>
Just	O
like	O
for	O
simple	O
graphs	O
,	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
of	O
a	O
directed	O
weighted	O
graph	O
is	O
by	O
definition	O
generally	O
non-symmetric	O
.	O
</s>
<s>
The	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
undirected	O
graph	O
could	O
,	O
e.g.	O
,	O
be	O
defined	O
as	O
a	O
sum	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
original	O
directed	O
graph	O
and	O
its	O
matrix	B-Architecture
transpose	O
as	O
in	O
the	O
following	O
example	O
:	O
</s>
<s>
where	O
the	O
zero	O
and	O
one	O
entries	O
of	O
are	O
treated	O
as	O
numerical	O
,	O
rather	O
than	O
logical	O
as	O
for	O
simple	O
graphs	O
,	O
values	O
,	O
explaining	O
the	O
difference	O
in	O
the	O
results	O
-	O
for	O
simple	O
graphs	O
,	O
the	O
symmetrized	O
graph	O
still	O
needs	O
to	O
be	O
simple	O
with	O
its	O
symmetrized	O
adjacency	B-Algorithm
matrix	I-Algorithm
having	O
only	O
logical	O
,	O
not	O
numerical	O
values	O
,	O
e.g.	O
,	O
the	O
logical	O
sum	O
is	O
1	O
v	O
1	O
=	O
1	O
,	O
while	O
the	O
numeric	O
sum	O
is	O
1	O
+	O
1	O
=	O
2	O
.	O
</s>
<s>
Alternatively	O
,	O
the	O
symmetric	B-Algorithm
Laplacian	B-Algorithm
matrix	I-Algorithm
can	O
be	O
calculated	O
from	O
the	O
two	O
Laplacians	O
using	O
the	O
indegree	O
and	O
outdegree	O
,	O
as	O
in	O
the	O
following	O
example	O
:	O
</s>
<s>
The	O
sum	O
of	O
the	O
out-degree	O
Laplacian	O
transposed	O
and	O
the	O
in-degree	O
Laplacian	O
equals	O
to	O
the	O
symmetric	B-Algorithm
Laplacian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
The	O
goal	O
of	O
normalization	O
is	O
,	O
like	O
for	O
simple	O
graphs	O
,	O
to	O
make	O
the	O
diagonal	O
entries	O
of	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
to	O
be	O
all	O
unit	O
,	O
also	O
scaling	O
off-diagonal	O
entries	O
correspondingly	O
.	O
</s>
<s>
Graph	O
self-loops	O
,	O
i.e.	O
,	O
non-zero	O
entries	O
on	O
the	O
main	O
diagonal	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
do	O
not	O
affect	O
the	O
graph	B-Algorithm
Laplacian	I-Algorithm
values	O
,	O
but	O
may	O
need	O
to	O
be	O
counted	O
for	O
calculation	O
of	O
the	O
normalization	O
factors	O
.	O
</s>
<s>
where	O
L	O
is	O
the	O
unnormalized	O
Laplacian	O
,	O
A	O
is	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
D	O
is	O
the	O
degree	B-Algorithm
matrix	I-Algorithm
,	O
and	O
is	O
the	O
Moore	O
–	O
Penrose	O
inverse	O
.	O
</s>
<s>
Since	O
the	O
degree	B-Algorithm
matrix	I-Algorithm
D	O
is	O
diagonal	O
,	O
its	O
reciprocal	O
square	O
root	O
is	O
just	O
the	O
diagonal	O
matrix	B-Architecture
whose	O
diagonal	O
entries	O
are	O
the	O
reciprocals	O
of	O
the	O
square	O
roots	O
of	O
the	O
diagonal	O
entries	O
of	O
D	O
.	O
If	O
all	O
the	O
edge	O
weights	O
are	O
nonnegative	O
then	O
all	O
the	O
degree	O
values	O
are	O
automatically	O
also	O
nonnegative	O
and	O
so	O
every	O
degree	O
value	O
has	O
a	O
unique	O
positive	O
square	O
root	O
.	O
</s>
<s>
The	O
symmetrically	O
normalized	O
Laplacian	O
is	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
if	O
and	O
only	O
if	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
A	O
is	O
symmetric	B-Algorithm
and	O
the	O
diagonal	O
entries	O
of	O
D	O
are	O
nonnegative	O
,	O
in	O
which	O
case	O
we	O
can	O
use	O
the	O
term	O
the	O
symmetric	B-Algorithm
normalized	O
Laplacian	O
.	O
</s>
<s>
using	O
the	O
weightless	O
incidence	B-Algorithm
matrix	I-Algorithm
B	O
and	O
the	O
diagonal	O
matrix	B-Architecture
W	O
containing	O
the	O
edge	O
weights	O
and	O
defining	O
the	O
new	O
weighted	O
incidence	B-Algorithm
matrix	I-Algorithm
whose	O
rows	O
are	O
indexed	O
by	O
the	O
vertices	O
and	O
whose	O
columns	O
are	O
indexed	O
by	O
the	O
edges	O
of	O
G	O
such	O
that	O
each	O
column	O
corresponding	O
to	O
an	O
edge	O
e	O
=	O
{	O
u	O
,	O
v}	O
has	O
an	O
entry	O
in	O
the	O
row	O
corresponding	O
to	O
u	O
,	O
an	O
entry	O
in	O
the	O
row	O
corresponding	O
to	O
v	O
,	O
and	O
has	O
0	O
entries	O
elsewhere	O
.	O
</s>
<s>
where	O
D	O
is	O
the	O
degree	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
Since	O
the	O
degree	B-Algorithm
matrix	I-Algorithm
D	O
is	O
diagonal	O
,	O
its	O
inverse	O
is	O
simply	O
defined	O
as	O
a	O
diagonal	O
matrix	B-Architecture
,	O
having	O
diagonal	O
entries	O
which	O
are	O
the	O
reciprocals	O
of	O
the	O
corresponding	O
diagonal	O
entries	O
of	O
D	O
.	O
For	O
the	O
isolated	O
vertices	O
(	O
those	O
with	O
degree	O
0	O
)	O
,	O
a	O
common	O
choice	O
is	O
to	O
set	O
the	O
corresponding	O
element	O
to	O
0	O
.	O
</s>
<s>
The	O
name	O
of	O
the	O
random-walk	O
normalized	O
Laplacian	O
comes	O
from	O
the	O
fact	O
that	O
this	O
matrix	B-Architecture
is	O
,	O
where	O
is	O
simply	O
the	O
transition	O
matrix	B-Architecture
of	O
a	O
random	O
walker	O
on	O
the	O
graph	O
,	O
assuming	O
non-negative	O
weights	O
.	O
</s>
<s>
The	O
random	O
walk	O
normalized	O
Laplacian	O
can	O
also	O
be	O
called	O
the	O
left	O
normalized	O
Laplacian	O
since	O
the	O
normalization	O
is	O
performed	O
by	O
multiplying	O
the	O
Laplacian	O
by	O
the	O
normalization	O
matrix	B-Architecture
on	O
the	O
left	O
.	O
</s>
<s>
It	O
has	O
each	O
row	O
summing	O
to	O
zero	O
since	O
is	O
right	B-Algorithm
stochastic	I-Algorithm
,	O
assuming	O
all	O
the	O
weights	O
are	O
non-negative	O
.	O
</s>
<s>
In	O
the	O
less	O
uncommonly	O
used	O
right	O
normalized	O
Laplacian	O
each	O
column	O
sums	O
to	O
zero	O
since	O
is	O
left	B-Algorithm
stochastic	I-Algorithm
.	O
</s>
<s>
For	O
a	O
non-symmetric	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
a	O
directed	O
graph	O
,	O
one	O
also	O
needs	O
to	O
choose	O
indegree	O
or	O
outdegree	O
for	O
normalization	O
:	O
</s>
<s>
The	O
left	O
out-degree	O
normalized	O
Laplacian	O
with	O
row-sums	O
all	O
0	O
relates	O
to	O
right	B-Algorithm
stochastic	I-Algorithm
,	O
while	O
the	O
right	O
in-degree	O
normalized	O
Laplacian	O
with	O
column-sums	O
all	O
0	O
contains	O
left	B-Algorithm
stochastic	I-Algorithm
.	O
</s>
<s>
Negative	O
weights	O
may	O
also	O
give	O
negative	O
row	O
-	O
and/or	O
column-sums	O
,	O
so	O
that	O
the	O
corresponding	O
diagonal	O
entry	O
in	O
the	O
non-normalized	O
Laplacian	B-Algorithm
matrix	I-Algorithm
would	O
be	O
negative	O
and	O
a	O
positive	O
square	O
root	O
needed	O
for	O
the	O
symmetric	B-Algorithm
normalization	O
would	O
not	O
exist	O
.	O
</s>
<s>
Arguments	O
can	O
be	O
made	O
to	O
take	O
the	O
absolute	O
value	O
of	O
the	O
row	O
-	O
and/or	O
column-sums	O
for	O
the	O
purpose	O
of	O
normalization	O
,	O
thus	O
treating	O
a	O
possible	O
value	O
-1	O
as	O
a	O
legitimate	O
unit	O
entry	O
of	O
the	O
main	O
diagonal	O
of	O
the	O
normalized	O
Laplacian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
For	O
an	O
(	O
undirected	O
)	O
graph	O
G	O
and	O
its	O
Laplacian	B-Algorithm
matrix	I-Algorithm
L	O
with	O
eigenvalues	O
:	O
</s>
<s>
L	O
is	O
symmetric	B-Algorithm
.	O
</s>
<s>
L	O
is	O
positive-semidefinite	B-Algorithm
(	O
that	O
is	O
for	O
all	O
)	O
.	O
</s>
<s>
This	O
can	O
be	O
seen	O
from	O
the	O
fact	O
that	O
the	O
Laplacian	O
is	O
symmetric	B-Algorithm
and	O
diagonally	O
dominant	O
.	O
</s>
<s>
L	O
is	O
an	O
M-matrix	B-Algorithm
(	O
its	O
off-diagonal	O
entries	O
are	O
nonpositive	O
,	O
yet	O
the	O
real	O
parts	O
of	O
its	O
eigenvalues	O
are	O
nonnegative	O
)	O
.	O
</s>
<s>
In	O
consequence	O
,	O
,	O
because	O
the	O
vector	O
satisfies	O
This	O
also	O
implies	O
that	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
is	O
singular	O
.	O
</s>
<s>
The	O
number	O
of	O
connected	O
components	O
in	O
the	O
graph	O
is	O
the	O
dimension	O
of	O
the	O
nullspace	B-Algorithm
of	O
the	O
Laplacian	O
and	O
the	O
algebraic	O
multiplicity	O
of	O
the	O
0	O
eigenvalue	O
.	O
</s>
<s>
When	O
G	O
is	O
k-regular	O
,	O
the	O
normalized	O
Laplacian	O
is	O
:	O
,	O
where	O
A	O
is	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
and	O
I	O
is	O
an	O
identity	O
matrix	B-Architecture
.	O
</s>
<s>
For	O
a	O
graph	O
with	O
multiple	O
connected	O
components	O
,	O
L	O
is	O
a	O
block	O
diagonal	O
matrix	B-Architecture
,	O
where	O
each	O
block	O
is	O
the	O
respective	O
Laplacian	B-Algorithm
matrix	I-Algorithm
for	O
each	O
component	O
,	O
possibly	O
after	O
reordering	O
the	O
vertices	O
(	O
i.e.	O
</s>
<s>
L	O
is	O
permutation-similar	O
to	O
a	O
block	O
diagonal	O
matrix	B-Architecture
)	O
.	O
</s>
<s>
The	O
trace	O
of	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
L	O
is	O
equal	O
to	O
where	O
is	O
the	O
number	O
of	O
edges	O
of	O
the	O
considered	O
graph	O
.	O
</s>
<s>
All	O
eigenvalues	O
of	O
the	O
normalized	O
symmetric	B-Algorithm
Laplacian	O
satisfy	O
0	O
=	O
μ0	O
≤	O
…	O
≤	O
μn−1	O
≤	O
2	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
even	O
if	O
is	O
in	O
general	O
not	O
symmetric	B-Algorithm
,	O
it	O
has	O
real	O
eigenvalues	O
—	O
exactly	O
the	O
same	O
as	O
the	O
eigenvalues	O
of	O
the	O
normalized	O
symmetric	B-Algorithm
Laplacian	O
.	O
</s>
<s>
The	O
graph	B-Algorithm
Laplacian	I-Algorithm
matrix	B-Architecture
can	O
be	O
further	O
viewed	O
as	O
a	O
matrix	B-Architecture
form	O
of	O
the	O
negative	O
discrete	B-Algorithm
Laplace	I-Algorithm
operator	I-Algorithm
on	O
a	O
graph	O
approximating	O
the	O
negative	O
continuous	O
Laplacian	O
operator	O
obtained	O
by	O
the	O
finite	B-Algorithm
difference	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
(	O
See	O
Discrete	B-Algorithm
Poisson	I-Algorithm
equation	I-Algorithm
)	O
In	O
this	O
interpretation	O
,	O
every	O
graph	O
vertex	O
is	O
treated	O
as	O
a	O
grid	O
point	O
;	O
the	O
local	O
connectivity	O
of	O
the	O
vertex	O
determines	O
the	O
finite	O
difference	O
approximation	O
stencil	B-Algorithm
at	O
this	O
grid	O
point	O
,	O
the	O
grid	O
size	O
is	O
always	O
one	O
for	O
every	O
edge	O
,	O
and	O
there	O
are	O
no	O
constraints	O
on	O
any	O
grid	O
points	O
,	O
which	O
corresponds	O
to	O
the	O
case	O
of	O
the	O
homogeneous	O
Neumann	O
boundary	O
condition	O
,	O
i.e.	O
,	O
free	O
boundary	O
.	O
</s>
<s>
Such	O
an	O
interpretation	O
allows	O
one	O
,	O
e.g.	O
,	O
generalizing	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
to	O
the	O
case	O
of	O
graphs	O
with	O
an	O
infinite	O
number	O
of	O
vertices	O
and	O
edges	O
,	O
leading	O
to	O
a	O
Laplacian	B-Algorithm
matrix	I-Algorithm
of	O
an	O
infinite	O
size	O
.	O
</s>
<s>
Entries	O
of	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
can	O
be	O
complex-valued	O
,	O
in	O
which	O
case	O
the	O
notion	O
of	O
the	O
matrix	B-Architecture
symmetry	O
needs	O
to	O
be	O
replaced	O
with	O
a	O
Hermitian	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
where	O
I	O
is	O
the	O
identity	O
matrix	B-Architecture
,	O
A	O
is	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
,	O
D	O
is	O
the	O
degree	B-Algorithm
matrix	I-Algorithm
,	O
and	O
s	O
is	O
a	O
(	O
complex-valued	O
)	O
number	O
.	O
</s>
<s>
where	O
is	O
the	O
degree	B-Algorithm
matrix	I-Algorithm
,	O
and	O
is	O
the	O
adjacency	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
where	O
is	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
An	O
analogue	O
of	O
the	O
Laplacian	B-Algorithm
matrix	I-Algorithm
can	O
be	O
defined	O
for	O
directed	O
multigraphs	O
.	O
</s>
<s>
where	O
D	O
is	O
a	O
diagonal	O
matrix	B-Architecture
with	O
Di	O
,	O
i	O
equal	O
to	O
the	O
outdegree	O
of	O
vertex	O
i	O
and	O
A	O
is	O
a	O
matrix	B-Architecture
with	O
Ai	O
,	O
j	O
equal	O
to	O
the	O
number	O
of	O
edges	O
from	O
i	O
to	O
j	O
(	O
including	O
loops	O
)	O
.	O
</s>
