Mapping of the symmetric vertex model onto the Ising model for an arbitrary lattice coordination

We investigate the mapping of the two-state symmetric vertex model onto the Ising model in a field for a lattice with general coordination number q. The analysis is based on a combination of decoration and generalized weak-graph transformations. It is shown that the mapping is restricted to the manifold formed by the intersection of q−3 hypersurfaces in the space of symmetric vertex weights. We also derive dual relations among the model parameters and the free energies of the related symmetric vertex and Ising models. The choices of the energies of the vertex configurations for which the mapping can be performed throughout the entire temperature range are briefly discussed.