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To a torus action on a compact oriented manifold $M$ with isolated fixedpoints, we associate a signed labeled multigraph encoding the fixed point data(weights and signs at fixed points and isotropy submanifolds) of the manifold.When the circle group acts on a 6-dimensional $M$, we classify such amultigraph by proving that we can convert it into the empty graph bysuccessively applying two types of operations. In particular, this calssifiesthe fixed point data of any such manifold. We prove this by showing that forany such manifold, we can successively take equivariant connected sums at fixedpoints with itself, $\mathbb{CP}^3$, and 6-dimensional analogue $Z_1$ and $Z_2$of the Hirzebruch surfaces (and these with opposite orientations) to a fixedpoint free action on a compact oriented 6-manifold. We also classify amultigraph for a torus action on a 4-dimensional $M$.

Graphs for torus actions on oriented manifolds with isolated fixed points and classification in dimension 6