Plans' periodicity theorem for Jacobian of circulant graphs

Plans' theorem states that, for odd n, the first homology group of the $n$-fold cyclic covering of the three-dimensional sphere branched over a knot is the direct product of two copies of an Abelian group. A similar statement holds for even $n$. In this case, one has to factorize the homology group of $n$-fold covering by the homology group of two-fold covering of the knot. The aim of this paper is to establish similar results for Jacobians (critical group) of a circulant graph. Moreover, it is shown that the Jacobian group of a circulant graph on $n$ vertices reduced modulo a given finite Abelian group is a periodic function of $n$.