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