This talk will provide basic principles of the Ising and q-state Potts model partition functions of statistical mechanics. These models play important roles in the theory of phase transitions and critical phenomena in physics, and have applications as widely varied as muscle cells, foam behaviors, and social demographics. The Potts model is constructed on various lattices, and when these lattices are viewed as graphs (i.e. networks of nodes and edges), then, remarkably, the Potts model is also equivalent to one of the most renown graph invariants, the Tutte polynomial. Thus, the talk will also give a general introduction to graph theory and the Tutte polynomial. The Tutte polynomial is the universal object of its type, in that any invariant that obeys a certain deletion/contraction relation (or equivalently a particular state-model formulation), must be an evaluation of it. The talk includes a (very) little history and some interesting properties of the Potts model partition function and Tutte polynomial, but the emphasis will be on how the Potts model and Tutte polynomial are related and how research into the one has informed the theory of the other, and vice versa. The talk concludes with locations of zeros corresponding to phase transitions and computational complexity analysis, and if time a brief excursion into Monte Carlo simulations. Applet from Peter Young at