GKM graph locally modeled by $T^n\times S^1$-action on $T^{\ast }{\mathbb{C}}^n$ and its graph equivariant cohomology

We introduce a class of labeled graphs (with legs) which contains two classes of GKM graphs of $4n$-dimensional manifolds with $T^n\times S^1$-actions, i.e., GKM graphs of the toric hyperK$\ddot{\mathrm{a}}$hler manifolds and of the cotangent bundles of toric manifolds. Under some conditions, the graph equivariant cohomology ring of such a labeled graph is computed. We also give a module basis of the graph equivariant cohomology by using a shelling structure of such a labeled graph and study their multiplicative structure.