  We consider the statics and dynamics of distinguishable spin-1/2 systems on
an arbitrary graph G with N vertices. In particular, we consider systems of
quantum spins evolving according to one of two hamiltonians: (i) the XY
hamiltonian H_XY, which contains an XY interaction for every pair of spins
connected by an edge in G; and (ii) the Heisenberg hamiltonian H_Heis, which
contains a Heisenberg interaction term for every pair of spins connected by an
edge in G. We find that the action of the XY (respectively, Heisenberg)
hamiltonian on state space is equivalent to the action of the adjacency matrix
(respectively, combinatorial laplacian) of a sequence G_k, k=0,..., N of graphs
derived from G (with G_1=G). This equivalence of actions demonstrates that the
dynamics of these two models is the same as the evolution of a free particle
hopping on the graphs G_k. Thus we show how to replace the complicated dynamics
of the original spin model with simpler dynamics on a more complicated
geometry. A simple corollary of our approach allows us to write an explicit
spectral decomposition of the XY model in a magnetic field on the path
consisting of N vertices. We also use our approach to utilise results from
spectral graph theory to solve new spin models: the XY model and heisenberg
model in a magnetic field on the complete graph.
