Optimal cuts in graphs and statistical mechanics

We survey well known problems from statistical mechanics involving optimal cuts of graphs. These problems include finding the ground states for the spin glass problem or for the random field Ising model, as well as finding the lowest energy barrier between the two ground states of a ferromagnet. The relations between the results in graph theory and in physics are outlined. In particular, the solvability of a special max cut problem which arises in statistical mechanics is an easy consequence of a gauge invariance. Throughout the paper, we review some useful algorithms and results. We also give a simple solution of the cutwidth problem in the case of a regular tree.