n ->infinity limit of O(n) ferromagnetic models on graphs

Thirty years ago, H.E. Stanley showed that an O(n) spin model on a lattice tends to a spherical model as n --> infinity. This means that at any temper ature the corresponding free energies coincide. This fundamental result is no longer valid on more general discrete structures lacking in translation invariance, i.e., on graphs. However, only the singular parts of the free e nergies determine the critical behavior of the two statistical models. Here we show that for ferromagnetic models such singular parts still coincide e ven on graphs in the thermodynamic limit. This implies that the critical ex ponents of O(n) models on graphs for n --> infinity tend to the spherical o nes and depend only on the graph spectral dimension.