  We provide an explicit formula for the limiting free energy density
(log-partition function divided by the number of vertices) for ferromagnetic
Potts models on uniformly sparse graph sequences converging locally to the
d-regular tree for d even, covering all temperature regimes. This formula
coincides with the Bethe free energy functional evaluated at a suitable fixed
point of the belief propagation recursion on the d-regular tree, the so-called
replica symmetric solution. For uniformly random d-regular graphs we further
show that the replica symmetric Bethe formula is an upper bound for the
asymptotic free energy for any model with permissive interactions.
