In the large-n limit of the O(n) model on graphs, free energy at low T is governed by the Laplacian spectrum and at high T by the adjacency spectrum.
Harary,Graph Theory(Addison-Wesley Publishing Company, Boston, 1969)
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From Laplacian-to-Adjacency Matrix for Continuous Spins on Graphs
In the large-n limit of the O(n) model on graphs, free energy at low T is governed by the Laplacian spectrum and at high T by the adjacency spectrum.