MHB Eigenvalues of Laplacian are non-negative

tarnat
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Hi, I need to learn the following proof and I'm having trouble getting my head round it. Any help would be appreciated.

Show that if vector x in R^n with components x=(x1,x2,...,xn), then
x.Lx=0.5 sum(Aij(xi-xj)^2)
where A is the graphs adjacency matrix, L is laplacian.
Then use this result to prove that the eigen values of L are non-zero for all 1<j<n.

Thanks.
 
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