Cohen-Macaulay Circulant Graphs

Let G be the circulant graph C_n(S) with S a subset of {1,2,…,\lfloor n/2 \rfloor}, and let I(G) denote its the edge ideal in the ring R = k[x_1,…,x_n]. We consider the problem of determining when G is Cohen-Macaulay, i.e, R/I(G) is a Cohen-Macaulay ring. Because a Cohen-Macaulay graph G must be well-covered, we focus on known families of well-covered circulant graphs of the form C_n(1,2,…,d). We also characterize which cubic circulant graphs are Cohen-Macaulay. We end with the observation that even though the well-covered property is preserved under lexicographical products of graphs, this is not true of the Cohen-Macaulay property.