  Recently, a class of interaction round the face (IRF) solvable lattice models
were introduced, based on any rational conformal field theory (RCFT). We
investigate here the connection between the general solvable IRF models and the
fusion ones. To this end, we introduce an associative algebra associated to any
graph, as the algebra of products of the eigenvalues of the incidence matrix.
If a model is based on an RCFT, its associated graph algebra is the fusion ring
of the RCFT. A number of examples are studied. The Gordon--generalized IRF
models are studied, and are shown to come from RCFT, by the graph algebra
construction. The IRF models based on the Dynkin diagrams of A-D-E are studied.
While the $A$ case stems from an RCFT, it is shown that the $D-E$ cases do not.
The graph algebras are constructed, and it is speculated that a natural
isomorphism relating these to RCFT exists. The question whether all solvable
IRF models stems from an RCFT remains open, though the $D-E$ cases shows that a
mixing of the primary fields is needed.
