One-skeleta, Betti numbers and equivariant cohomology

The one-skeleton of a G-manifold M is the set of points p in M where $\dim G_p \geq \dim G -1$; and M is a GKM manifold if the dimension of this one-skeleton is 2. Goresky, Kottwitz and MacPherson show that for such a manifold this one-skeleton has the structure of a ``labeled" graph, $(\Gamma, \alpha)$, and that the equivariant cohomology ring of M is isomorphic to the ``cohomology ring'' of this graph. Hence, if M is symplectic, one can show that this ring is a free module over the symmetric algebra $\SS(\fg^*)$, with $b_{2i}(\Gamma)$ generators in dimension 2i, $b_{2i}(\Gamma)$ being the ``combinatorial'' 2i-th Betti number of $\Gamma$. In this article we show that this ``topological'' result is , in fact, a combinatorial result about graphs.