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

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