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Unformatted text preview: 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 self-loops. 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 otherwise Since L is symmetric, we can call its eigenvalues 1 ... 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...
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This note was uploaded on 09/30/2011 for the course MATH 262 taught by Professor Aterras during the Fall '08 term at UCSD.
- Fall '08