Geodesic
The hard Lefschetz property for Hamiltonian GKM manifolds
Journal of Algebraic Combinatorics, Tome 40 (2014) no. 1, pp. 45-74.

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We introduce characteristic numbers of a graph and demonstrate that they are a combinatorial analogue of topological Betti numbers. We then use characteristic numbers and related tools to study Hamiltonian GKM manifolds whose moment maps are in general position. We study the connectivity properties of GKM graphs and give an upper bound on the second Betti number of a GKM manifold. When the manifold has dimension at most 10, we use this bound to conclude that the manifold has nondecreasing even Betti numbers up to half the dimension, which is a weak version of the hard Lefschetz property.
Classification : 05C10, 57S10, 22D35, 57M27
Keywords: graph cohomology, hard Lefschetz property, Hamiltonian GKM 
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Luo, Shisen. The hard Lefschetz property for Hamiltonian GKM manifolds. Journal of Algebraic Combinatorics, Tome 40 (2014) no. 1, pp. 45-74. http://geodesic.mathdoc.fr/item/JAC_2014__40_1_a10/