Abstract
We prove discrete versions of nodal domain theorems; in
particular, an eigenvector corresponding to the $s$th 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.