Let $X$ be a possibly singular projective scheme which admits a torus $T$ action and has finitely many $T$ fixed points and one-dimensional $T-$orbits. There are many such schemes in the Grassmannian/flag variety for an algebraic group/Kac-Moody group. Then some theorems by GKM(Goresky-Kottwitz-MacPherson) allow us to compute the (equivariant)cohomology of $X$ from the moment graph of $X$, which consists of the zero and one dimensional $T-$orbits.
Based on the Bialynicki-Birula Decomposition (or equivalently on Morse Theory), we can give $X$ a "cell decomposition", and the cells are the "attracting sets" of the fixed points. My questions are: how is the information of this cell decomposition reflected in the moment graph defined above? In particular, given a $T-$fixed point $p$, how is its attracting set/cell related to the set of one-dimensional edges connected to $p$ in the moment graph for $X$? For example, can we get a bound on the dimension of this cell by counting the number of the one-dimensional edges connected to $p$ in the moment graph? Any references would be appreciated. Thanks!