The simplest nearest-neighbor spin system on regular graphs:Time   dynamics of the mean coverage function
  We establish a characterization of the class of the simplest nearest neighbor spin systems possesing the mean coverage function (mcf) that obeys a second order differential equation, and derive explicit expressions for the mcf's of the above models. Based on these expressions, the problem of ergodicity of the models is studied and bounds for their spectral gaps are obtained. 
Boris L. Granovsky