The graphical regular representations of finite metacyclic p-groups

A Cayley graph Gamma = Cay(G, S) is called a graphical regular representation of the group G if Aut Gamma = G. One long-standing open problem about Cayley graphs is to determine which Cayley graphs are graphical regular representations of the corresponding groups. A simple necessary condition for Gamma to be a graphical regular representation of G is Aut(G, S) = 1, where Aut(G, S) = (tau is an element of Aut(G) \ S-tau = S). C. Godsil in (Europ. J. Combinatories, 4 (1983)) proposed to characterize graphical regular representations of groups C; in terms of Aut(G, S); that is, for a given class of groups G, find the conditions under which Cay(G, S) is a graphical regular representation of G if and only if Aut(G, S) = 1. The main purpose of this paper is to give a complete solution to this problem for the class of metacyclic p-groups where p is a prime. (C) 2000 Academic Press.