YuhJie (Eunice) Chen
Dirichlet Eigenvalues
1 Introduction
In this lecture, we will go over the basics of the Dirichlet eigenvalues and prove a matrixtree
theorem.
2 Dirichlet eigenvalues
2.1 Boundary
The idea of boundary pervades many divisions of mathematics, such as numerical method and
differential geometry. Not surprisingly, it also enters the realm of graph theory.
In a graph
G
(
V,E
)
,
for a set
S
⊆
V,
the induced subgraph determined by
S
has edge set consisting
of all edges in
E
that have both endpoints in
S.
In this lecture note, when there is no confusion,
we will simply denote by
S
the induced subgraph determined by
S.
For an induced graph
S
, there are two kinds of boundaries: the vertex boundary
δS
and the edge
boundary
∂S
. The vertex boundary is deﬁned as
δS
=
{
u
∈
V
:
u /
∈
S,u
∼
v
∈
S
}
.
The edge boundary is deﬁned as
∂S
=
±
{
u,v
} ∈
E
:
u
∈
S,v /
∈
S
²
.
2.2 Dirichlet boundary condition
Assume
δS
6
=
∅
,
the Dirichlet boundary condition is stated as follows:
for all
x
∈
δS, f
(
x
) = 0
,
where function
f
:
V
→
R
.
1
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 Fall '08
 aterras
 Math, Graph Theory, Glossary of graph theory, Dirichlet eigenvalues

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