Graph Laplacians, Nodal Domains, and Hyperplane Arrangements

Eigenvectors of the Laplacian of a graph G have received increasing attention in the recent past. Here we investigate their so-called nodal domains, i.e., the connected components of the maximal induced subgraphs of G on which an eigenvector \psi does not change sign. An analogue of Courant's nodal domain theorem provides upper bounds on the number of nodal domains depending on the location of \psi in the spectrum. This bound, however, is not sharp in general. In this contribution we consider the problem of computing minimal and maximal numbers of nodal domains for a particular graph. The class of Boolean Hypercubes is discussed in detail. We find that, despite the simplicity of this graph class, for which complete spectral information is available, the computations are still non-trivial. Nevertheless, we obtained some new results and a number of conjectures.