  We study XXZ spin systems on general graphs. In particular, we describe the
formation of droplet states near the bottom of the spectrum in the Ising phase
of the model, where the Z-term dominates the XX-term. As key tools we use
particle number conservation of XXZ systems and symmetric products of graphs
with their associated adjacency matrices and Laplacians. Of particular interest
to us are strips and multi-dimensional Euclidean lattices, for which we discuss
the existence of spectral gaps above the droplet regime. We also prove a
Combes-Thomas bound which shows that the eigenstates in the droplet regime are
exponentially small perturbations of strict (classical) droplets.
