Symmetry properties of generalized graph truncations

In the generalized truncation construction, one replaces each vertex of a k-regular graph Γ with a copy of a graph ϒ of order k. We investigate the symmetry properties of the graphs constructed in this way, especially in connection to the symmetry properties of the graphs Γ and ϒ used in the construction. We demonstrate the usefulness of our results by using them to obtain a classification of cubic vertex-transitive graphs of girths 3, 4, and 5.