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In this paper we propose a spectral flow for graph Laplacians, and prove thatit counts the number of nodal domains for a given Laplace eigenvector. Thisextends work done for Laplacians on $\mathbb{R}^n$ to the graph setting. Wemention some open problems relating the topology of a graph to the analyticbehaviour of its Laplace eigenvectors, and include numerical examplesillustrating our flow.

A Graph Spectral Flow for Computing Nodal Deficiencies