On graphical regular representations of cyclic extensions of groups

Wilfried Imrich and Mark E. Watkins

Vol. 55 (1974), No. 2, 461–477

DOI: 10.2140/pjm.1974.55.461

Abstract

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 : G] = 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.

Mathematical Subject Classification 2000

Primary: 05C25

Milestones

Received: 2 February 1972

Revised: 1 October 1973

Published: 1 December 1974