Some algebraic invariants of edge ideal of circulant graphs

Let G be the circulant graph C-n(S) with S subset of {1,...,[n/2]} and let I(G) be its edge ideal in the ring K[x(0),...,x(n-1)]. Under the hypothesis that n is prime we : 1) compute the regularity index of R/I(G); 2) compute the Castelnuovo-Mumford regularity when R/I(G) is Cohen-Macaulay; 3) prove that the circulant graphs with S = {1,...,s} are sequentially S-2. We end characterizing the Cohen-Macaulay circulant graphs of Krull dimension 2 and computing their Cohen-Macaulay type and Castelnuovo-Mumford regularity.