We consider a family of graphs generalizing the family of $ I $ -graphs, which in turn includes generalized Petersen graphs and prismatic graphs.The paper is devoted to the study of the critical group of a graph that is a cone over a generalized $ I $ -graph.The main result of the article is an analog of the Plans theorem (1953), which describes the first homology group of an $ n $ -sheeted cyclic cover of the three-dimensional sphere branched over a knot.It asserts that this homology group is almost a direct sum of two copies of a certain abelian group.In this paper, analogous results are established for the structure of the critical group of the graphs under consideration.
