This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 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 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...
View
Full
Document
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
 aterras
 Math

Click to edit the document details