  We present a complete analytical solution of a system of Potts spins on a
random k-regular graph in both the canonical and microcanonical ensembles,
using the Large Deviation Cavity Method (LDCM). The solution is shown to be
composed of three different branches, resulting in an non-concave entropy
function.The analytical solution is confirmed with numerical Metropolis and
Creutz simulations and our results clearly demonstrate the presence of a region
with negative specific heat and, consequently, ensemble inequivalence between
the canonical and microcanonical ensembles.
