For a graph G, Bayer–Denker–Milutinović–Rowlands–Sundaram–Xue study in [1] a new graph complex \(\Delta _k^t(G)\) , 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.
