On graphical regular representations of cyclic extensions of groups
Wilfried Imrich, Mark E. Watkins · Pacific Journal of Mathematics · 1974
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 Gι is a cyclic extension of a group G which admits a GRR, the question is posed whether G 1 also admits a GRR.Nowitz and Watkins have given an affirmative answer if Gt is non-abelian and finite and the index [Gχi G] ^ 5.This paper applies some new graph theoretical techniques to investigate the problem if [G t : G] = 2, 3 or 4, whether or not Gι is finite.As long as G x is non-abelian, an affirmative answer can again be given except in only finitely many unresolved cases.