Vol. 55, No. 2, 1974

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On graphical regular representations of cyclic extensions of groups

Wilfried Imrich and Mark E. Watkins

Vol. 55 (1974), No. 2, 461–477

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
Received: 2 February 1972
Revised: 1 October 1973
Published: 1 December 1974
Wilfried Imrich
Montanuniversität Leoben
Franz-Josef-Straße 18
8700 Leoben
Mark E. Watkins