  In this work we show that there is a direct relationship between a graph's
topology and the free energy of a spin system on the graph. We develop a method
of separating topological and enthalpic contributions to the free energy, and
find that considering the topology is sufficient to qualitatively compare the
free energies of different graph systems at high temperature, even when the
energetics are not fully known. This method was applied to the metal lattice
system with defects, and we found that it partially explains why point defects
are more stable than high-dimensional defects. Given the energetics, we can
even quantitatively compare free energies of different graph structures via a
closed form of linear graph contributions. The closed form is applied to
predict the sequence space free energy of lattice proteins, which is a key
factor determining the designability of a protein structure.
