For a connected graph G, the modified logarithmic vector field D_R(A_G) is isomorphic as an R-module to the face ring K[P_G] of an explicitly constructed simplicial poset, yielding new formulas for Hilbert series, local cohomology, projective dimension, and regularity.
Duval, Free resolutions of simplicial posets
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Vector fields of graphic arrangements and face rings of simplicial posets
For a connected graph G, the modified logarithmic vector field D_R(A_G) is isomorphic as an R-module to the face ring K[P_G] of an explicitly constructed simplicial poset, yielding new formulas for Hilbert series, local cohomology, projective dimension, and regularity.