Edge ideals and DG algebra resolutions

Let R = S/I where S = k [T-1, ... ,T-n] and I is a homogeneous ideal in S. The acyclic closure R < Y > of k over R is a DG algebra resolution obtained by means of Tate's process of adjoining variables to kill cycles. In a similar way one can obtain the minimal model S [X], a DG algebra resolution of R over S. By a theorem of Avramov there is a tight connection between these two resolutions. In this paper we study these two resolutions when I is the edge ideal of a path or a cycle. We determine the behavior of the deviations epsilon(i) ( R), which are the number of variables in R < Y > in homological degree i. We apply our results to the study of the k-algebra structure of the Koszul homology of R.