The other eigenvectors of the Laplacian

We are now going to begin our study of the other eigenvalues and eigenvectors of the Laplacian. I will begin the lecture by showing how much of the theory we established can be preserved. We will then determine the eigenvalues of the hypercube, and begin to see why λ 2 is so important. Recall that we showed that the kth eigenvector of a path graph crosses the origin at most k − 1 times. For example, here are the first three non-constant eigenvectors of the path graph on 13 vertices, with a line drawn at the origin. Today, we will prove a result of Fiedler [Fie75] which says that for every G the graph induced on the vertices that are non-negative in the kth eigenvector has at most k − 1 connected components. In particular, it says that the non-negative vertices in v 2 are connected. First, we need to recall a little linear algebra. In Lecture 3, we proved the Perron-Frobenius Theorem for non-negative matrices. I wish to quickly observe that this theory may also be applied to Laplacian matrices, to principal sub-matrices of Laplacian matrices, and to any matrix with non-positive off-diagonal entries. The difference is that it then involves the eigenvector of the smallest eigenvalue, rather than the largest eigenvalue. Corollary 5.3.1. Let M be a matrix with non-positive off-diagonal entries, such that the graph of the non-zero off-diagonally entries is connected. Let λ 1 be the smallest eigenvalue of M and let v 1 be the corresponding eigenvector. Then v 1 may be taken to be strictly positive, and λ 1 has multiplicity 1.