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Lecture4

# Lecture4 - Lecture 3 Eigenvalues of the Laplacian...

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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 0 otherwise Since L is symmetric, we can call its eigenvalues λ 0 λ 1 . . . λ 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 B from the proof of the Matrix Tree

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Lecture4 - Lecture 3 Eigenvalues of the Laplacian...

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