Some remarks on symmetrical extensions of graphs

It is proved that results from a previous paper of the author on symmetrical 2-extensions of graphs can be extended to symmetrical p-extensions of graphs for any prime p. In particular, it is proved that, for any prime p, there are only finitely many symmetrical p-extensions of a locally finite graph with an abelian subgroup of finite index in its automorphism group. Some refinements of these results are also obtained. In addition, we consider the question on the possibility to represent symmetrical extensions of a d-dimensional grid (and similar graphs) in the d-dimensional affine Euclidean space in such a way that a corresponding vertex-transitive group of automorphisms of the extension is induced by some crystallographic group of motions of the space.