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For a simplicial complex $\Delta$ï¿½??, ï¿½??the affect of the expansion functor on combinatorial properties of $\Delta$ and algebraic properties of its Stanley-Reisner ring has been studied in some previous papersï¿½??.
ï¿½??In this paperï¿½??, ï¿½??we consider the facet ideal $I(\Delta)$ and its Alexander dual which we denote by $J_{\Delta}$ to see how the expansion functor alter the algebraic properties of these idealsï¿½??. ï¿½??It is shown that for any expansion $\Delta^{\alpha}$ the ideals $J_{\Delta}$ and $J_{\Delta^{\alpha}}$ have the same total Betti numbers and their Cohen-Macaulayness are equivalentï¿½??, ï¿½??which implies that the regularities of the ideals $I(\Delta)$ and $I(\Delta^{\alpha})$ are equalï¿½??. ï¿½??Moreoverï¿½??, ï¿½??the projective dimensions of $I(\Delta)$ and $I(\Delta^{\alpha})$ are comparedï¿½??.
ï¿½??In the sequel for a graph $G$ï¿½??, ï¿½??some properties that are equivalent in $G$ and its expansions are presented and for a Cohen-Macaulay (respï¿½??. ï¿½??sequentially Cohen-Macaulay and shellable) graph $G$ï¿½??, ï¿½??we give some conditions for adding or removing a vertex from $G$ï¿½??, ï¿½??so that the remaining graph is still Cohen-Macaulay (respï¿½??. ï¿½??sequentially Cohen-Macaulay and shellable).
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