On Stanley-Reisner rings with linear resolution

For a graph $G$, Bayer-Denker-Milutinović-Rowlands-Sundaram-Xue study in \cite{B-D-M-R-S-X} a new graph complex ${\mathrm{\Delta }}_k^t(G)$, namely the simplicial complex with facets that are complements to independent sets of size $k$ in $G$. They are interested in topological properties such as shellability, vertex decomposability, homotopy type, and homology of these complexes. In this paper we study more algebraic properties, such as Cohen-Macaulayness, Betti numbers, and linear resolutions of the Stanley-Reisner ring of these complexes and their Alexander duals.