On graphical regular representations of cyclic extensions of groups

A simple graph X is said to be a graphical regular representation (GRR) of an abstract group G if the automorphism group of X is a regular permutation group and is isomorphic to G. If a group G1 is a cyclic extension of a group G which admits a GRR, the question is posed whether G1 also admits a GRR. Nowitz and Watkins have given an affirmative answer if G1 is non-abelian and finite and the index [G1: G] 5. This paper applies some new graph theoretical techniques to investigate the problem if [G1: ≧] = 2, 3 or 4, whether or not G1 is finite. As long as G1 is non-abelian, an affirmative answer can again be given except in only finitely many unresolved cases.