The bivariate Ising polynomial of a graph

In this paper we discuss the two variable Ising polynomials in a graph theoretical setting. This polynomial has its origin in physics as the partition function of the Ising model with an external field. We prove some basic properties of the Ising polynomial and demonstrate that it encodes a large amount of combinatorial information about a graph. We also give examples which prove that certain properties, such as the chromatic number, are not determined by the Ising polynomial. Finally we prove that there exist large families of non-isomorphic planar triangulations with identical Ising polynomial.