GKM Graph Locally Modeled by \(T^{n}\times S^{1}\) -Action on \(T^{*}\mathbb {C}^{n}\) 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 hyperKähler 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.