  Thirty years ago, Stanley showed that an O(n) spin model on a lattice tends
to a spherical model as $n\to\infty$. This means that at any temperature the
corresponding free energies coincide. This fundamental result, providing the
basis for more detailed studies of continuous symmetry spin models, is no
longer valid on more general discrete structures lacking of translation
invariance, i.e. on graphs. However only the singular parts of the free
energies determine the critical behavior of the two statistical models. Here we
show that such singular parts still coincide even on general graphs in the
thermodynamic limit. This implies that the critical exponents of O(n) models on
graphs for $n\to\infty$ tend to the spherical ones and therefore they only
depend on the graph spectral dimension.
