  In this paper we propose a spectral flow for graph Laplacians, and prove that
it counts the number of nodal domains for a given Laplace eigenvector. This
extends work done for Laplacians on $\mathbb{R}^n$ to the graph setting. We
mention some open problems relating the topology of a graph to the analytic
behaviour of its Laplace eigenvectors, and include numerical examples
illustrating our flow.
