Let G be a connected graph with Laplacian eigenvalues 0 = λ1 < λ2 ≤ ... ≤ λn. Which statement is always true about λ2?
10pts
Which of the following graphs has Laplacian eigenvalues 0, 1, 1, ..., 1, n (with 1 repeated n-2 times)?
10pts
For a simple graph G with Laplacian matrix L, which vector is always in the kernel of L?
10pts
Which theorem gives a variational characterization of Laplacian eigenvalues using the Rayleigh quotient?
10pts
If G is a connected graph and λ2 is its second-smallest Laplacian eigenvalue, which graph property is most directly controlled by λ2?
10pts