Perron–Frobenius type results and discrete versions of nodal domain theorems

Abstract

We prove discrete versions of nodal domain theorems; in particular, an eigenvector corresponding to the sth smallest eigenvalue of a graph Laplacian has at most s nodal domains. We compare our results to those of Courant and Pleijel on nodal domains of continuous Laplacians, and to those of Fiedler on nonnegative regions of graph Laplacians.