The Graphical Regular Representations of Finite Metacyclicp-Groups

A Cayley graph Γ = Cay(G, S) is called a graphical regular representation of the group G if AutΓ = 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 Γ to be a graphical regular representation of G isAut (G, S) = 1, where Aut(G, S) = { τ ∈ Aut(G) |Sτ = S }. C. Godsil in (Europ. J. Combinatorics, 4 (1983)) proposed to characterize graphical regular representations of groups G in terms ofAut (G, S); that is, for a given class of groups G, find the conditions under whichCay (G, S) is a graphical regular representation of G if and only ifAut (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.