Associated to a simple undirected graph G is a simplicial complex ∆G whose faces correspond to the independent sets of G. We call a graph G shellable if ∆G is a shellable simplicial complex in the non-pure sense of Björner-Wachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we cla...

We prove that sequentially Cohen-Macaulay rings in positive characteristic, as well as sequentially Cohen-Macaulay Stanley-Reisner rings in any characteristic, have trivial Lyubeznik table. Some other configurations of Lyubeznik tables are also provided depending on the deficiency modules of the ring.

We prove a Cohen-Macaulay version of result by Avramov-Golod and Frankild-J{\o}rgensen about Gorenstein rings, showing that if noetherian ring $A$ is Cohen-Macaulay, $a_1,\dots,a_n$ any sequence elements in $A$, then the Koszul complex $K(A;a_1,\dots,a_n)$ DG-ring. further generalize this result, it also holds for commutative DG-rings. In process proving this, we develop new technique to study ...