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. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Let G be a connected graph with n vertices and let x=(x1, ..., xn) be a real vector. A positive (negative) sign graph of the vector x is a maximal connected subgraph of G on vertices xi>0 (xi<0). For an eigenvalue of a generalized Laplacian of a tree: We characterize the maximal number of sign graphs of an eigenvector. We give an O(n2) time algorithm to find an eigenvector with maximum number of sign graphs and we show that finding an eigenvector with minimum number of sign graphs is an NP-complete problem. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing