  In this work we show that the simple Hamiltonians used in Quantum Graphity
models are highly degenerate, having multiple ground states that are not
lattices. In order to assess the distance of the resulting graphs from a
lattice graph, we propose a new measure of the equivalence of vertices in the
graph. We then propose a Hamiltonian that has a rectangular lattice as a ground
state that appears to be non-degenerate. The resulting graphs are close to
being a rectangular lattice, and the defects from the perfect lattice seem to
behave like particles of quantized mass that attract one another.
