We compute the Castelnuovo-Mumford regularity of the edge ideals of two families of circulant graphs, which includes all cubic circulant graphs. A feature of our approach is to combine bounds on the regularity, the projective dimension, and the reduced Euler characteristic to derive an exact value for the regularity.2000 Mathematics Subject Classification. 13D02,05C25,13F55.

Let K be a finite field. Let X* be a subset of the a ne space Kn, which is parameterized by odd cycles. In this paper we give an explicit Gröbner basis for the vanishing ideal, I(X*), of X*. We give an explicit formula for the regularity of I(X*) and finally if X* is parameterized by an odd cycle of length k, we show that the Hilbert function of the vanishing ideal of X* can be written as linear combination of Hilbert functions of degenerate torus.