  We present a mapping of dynamical graphs and, in particular, the graphs used
in the Quantum Graphity models for emergent geometry, into an Ising hamiltonian
on the line graph of a complete graph with a fixed number of vertices. We use
this method to study the properties of Quantum Graphity models at low
temperature in the limit in which the valence coupling constant of the model is
much greater than the coupling constants of the loop terms. Using mean field
theory we find that an order parameter for the model is the average valence of
the graph. We calculate the equilibrium distribution for the valence as an
implicit function of the temperature. In the approximation in which the
temperature is low, we find the first two Taylor coefficients of the valence in
the temperature expansion. A discussion of the susceptibility function and a
generalization of the model are given in the end.
