  In this paper, we study the hard Lefschetz property of a symplectic manifold
which admits a Hamiltonian torus action. More precisely, let $(M,\omega)$ be a
6-dimensional compact symplectic manifold with a Hamiltonian $T^2$-action. We
will show that if the moment map image of $M$ is a GKM-graph and if the graph
is index-increasing, then $(M,\omega)$ satisfies the hard Lefschetz property.
