Cohen-Macaulayness in Codimension for Line Simplicial Complexes

Abstract

Cohen-Macaulay complexes were first proposed in the mid-1970s. As the primary devel- oper of these inventions, Stanley has made major contributions over a long period of time. The idea of CMt simplicial complexes was itroduced by Nahandi, Yassemi, and Haghighi in 2012. We show that cycle graph having j vertices is unmixed for odd j 3 and it is not unmixed for even j 6 and prism graph having Jj vertices with J 3, j 2 is unmixed for odd J 3 and it is not unmixed for even J 6. Furthermore, we prove that the line simplicial complexes associated to cycle graph having j vertices are unmixed for odd j 3 and it is not unmixed for even j 6 and line simplicial complexes associated to prism graph YJ,2 with J 3 is unmixed. Also, we show that the line simplicial complexes associated to ladder graph having 2n vertices is Cohen-Macaulay and consequently it is CMt, for t 0. We provide example that DL(C5) is not CM0 but it is CMt, for t 1 .