For a simple undirected graph G with Laplacian matrix L, which statement best characterizes the multiplicity of the eigenvalue 0?
10pts
Let G be a connected graph with Laplacian eigenvalues 0 = λ1 < λ2 ≤ ... ≤ λn. What is the standard name of λ2?
10pts
Which condition on the Laplacian spectrum is equivalent to a graph being connected?
10pts
For a disconnected graph with exactly k connected components, what can be said about the Laplacian spectrum near 0?
10pts
Which vector is always an eigenvector of the Laplacian L corresponding to eigenvalue 0 for any graph G?
10pts
If a graph G is connected, what does the Rayleigh quotient x^T L x / x^T x reveal about λ2 when x is orthogonal to the all-ones vector?
10pts
Which of the following is true about the Laplacian matrix L of any simple undirected graph?
10pts
For a connected graph, which statement best interprets a very small λ2?
10pts