  In a series of papers we have found identical behaviour for various spin
models on thin random graphs - Feynman diagrams - and the corresponding Bethe
lattices. In this note we observe that in all cases the ratios of various
saddle point equations in the random graph approach are identical in form to
the fixed point(s) of the recursion relations which are used to solve the
models on the Bethe lattice. The loops in the random graphs thus have no
influence in the thermodynamic limit for such ferromagnetic spin models.
  We consider the correspondence explicitly for Ising and q state Potts models
and also note that multi spin interaction models on cacti admit a similar
correspondence with a randomised version of the cacti graphs which contain
loops.
