Kac-Ward Formula and Its Extension to Order-Disorder Correlators Through a Graph Zeta Function

A streamlined derivation of the Kac-Ward formula for the planar Ising model's partition function is presented and applied in relating the kernel of the Kac-Ward matrices' inverse with the correlation functions of the Ising model's order-disorder correlation functions. A shortcut for both is facilitated by the Bowen-Lanford graph zeta function relation. The Kac-Ward relation is also extended here to produce a family of non planar interactions on Z(2) for which the partition function and the order-disorder correlators are solvable at special values of the coupling parameters/temperature.