مشخصات پژوهش

For a simplicial complex , the effect of the expansion functor on combinatorial properties ofand algebraic properties of its Stanley–Reisner ring has been studied in some previous papers. In this paper, we consider the facet ideal I () and its Alexander dual which we denote by J to see how the expansion functor alters the algebraic properties of these ideals. It is shown that for any expansion α the ideals J and Jα have the same total Betti numbers and their Cohen–Macaulayness is equivalent, which implies that the regularities of the ideals I () and I(α) are equal. Moreover, the projective dimensions of I () and I(α) are compared. In the sequel for a graph G, some properties that are equivalent in G and its expansions are presented and for a Cohen–Macaulay (respectively, sequentially Cohen–Macaulay and shellable) graph G, we give some conditions for adding or removing a vertex from G, so that the remaining graph is still Cohen–Macaulay (respectively, sequentially Cohen–Macaulay and shellable).