AbstractBeck et al. characterized the grid graphs whose perfect matching polytopes areGorenstein and they also showed that for some parameters, perfect matching poly-topes of torus graphs are Gorenstein. In this paper, we complement their result,that is, we characterize the torus graphs whose perfect matching polytopes areGorenstein. Beck et al. also gave a method to construct an infinite family of Goren-stein polytopes. In this paper, we introduce a new class of polytopes obtained fromgraphs and we extend their method to construct many more Gorenstein polytopes...