The one-skeleton of a G-manifold M is the set of points p ∈ M where dim Gp ≥ 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, (Γ, α), 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 S(g * ), with b 2i (Γ) generators in dimension 2i, b 2i (Γ) being the "combinatorial" 2i-th Betti number of Γ. In this article we show that this "topological" result is , in fact, a combinatorial result about graphs.

Let T be a torus of dimension n > 1 and M a compact T -manifold. M is a GKM manifold if the set of zero dimensional orbits in the orbit space M/T is zero dimensional and the set of one dimensional orbits in M/T is one dimensional. For such a manifold these sets of orbits have the structure of a labelled graph and it is known that a lot of topological information about M is encoded in this graph.In this paper we prove that every compact homogeneous space M of non-zero Euler characteristic is of GKM type and show that the graph associated with M encodes geometric information about M as well as topological information. For example, from this graph one can detect whether M admits an invariant complex structure or an invariant almost complex structure.

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Abstract. The equivariant cohomology ring of a GKM manifold is isomorphic to the cohomology ring of its GKM graph. In this paper we explore the implications of this fact for equivariant fiber bundles for which the total space and the base space are both GKM and derive a graph theoretical version of the Leray-Hirsch theorem. Then we apply this result to the equivariant cohomology theory of flag varieties.