1-skeleta, Betti numbers, and equivariant cohomology

The 1-skeleton of a G-manifold M is the set of points p epsilon M, where di m G(p) greater than or equal to dim G - 1, and M is a GKM manifold if the d imension of this 1-skeleton is 2. M. Goresky, R. Kottwitz, and R. MacPherso n show that for such a manifold this 1-skeleton has the structure of a "lab eled" graph, (Gamma, alpha), and that the equivariant cohomology ring of M is isomorphic to the "cohomology ring" of this graph. Hence, if M is symple ctic, one can show that this ring is af ree module over the symmetric algeb ra S(g*), 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 th at this "topological" result is, in fact, a combinatorial result about grap hs.