Linear criteria for lifting automorphisms of elementary abelian regular coverings

Abstract

For a given finite connected graph Γ, a group H of automorphisms of Γ and a finite group A, a natural question can be raised as follows: Find all the connected regular coverings of Γ having A as its covering transformation group, on which each automorphism in H can be lifted. In this paper, we investigate the regular coverings with A= Z p n , an elementary abelian group and get some new matrix-theoretical characterizations for an automorphism of the base graph to be lifted. As one of its applications, we classify all the connected regular covering graphs of the Petersen graph satisfying the following two properties: (1) the covering transformation group is isomorphic to the elementary abelian p-group Z p n , and (2) the group of fibre-preserving automorphisms of a covering graph acts arc-transitively. As a byproduct, some new 2- and 3-arc-transitive graphs are constructed.