  In this paper we study the Reidemeister spectrum of 2-step nilpotent groups
associated to graphs. We develop three methods, based on the structure of the
graph, that can be used to determine the Reidemeister spectrum of the
associated group in terms of the Reidemeister spectra of groups associated to
smaller graphs. We illustrate our methods for several families of graphs,
including all the groups associated to a graph with at most four vertices. We
also apply our results in the context of topological fixed point theory for
nilmanifolds.
