Eigenfunctions and Nodal Domains

In the previous chapter we have seen that (due to the Perron-Frobenius Theorem) the eigenfunctions of the first eigenvalue λ1 have all entries positive (or negative) for a generalized Laplacian matrix M of a connected graph G. Fiedler [67] has shown that for eigenfunctions of the smallest nonzero eigenvalue of a graph the subgraph induced by nonpositive vertices (i.e., vertices with nonpositive function values) and the subgraph induced by nonnegative vertices are both connected. In other words, an eigenfunction of the second eigenvalue has exactly two weak nodal domains (also called weak sign graphs).