  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 \emph{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.
