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

Graph's Topology and Free Energy of a Spin Model on the Graph