Lecture6

Lecture6 - April 14, 2010 Yuh-Jie (Eunice) Chen Dirichlet...

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Yuh-Jie (Eunice) Chen Dirichlet Eigenvalues 1 Introduction In this lecture, we will go over the basics of the Dirichlet eigenvalues and prove a matrix-tree 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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Lecture6 - April 14, 2010 Yuh-Jie (Eunice) Chen Dirichlet...

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