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 $\mathbb{Z}^2$ for which the partition function and the order-disorder correlators are solvable at special values of the coupling parameters/temperature.