Graphical Regular Representations of Non-Abelian Groups, I

Abstract
In this paper, all groups and graphs considered are finite and all graphs are simple (in the sense of Tutte [8, p. 50]). IfXis such a graph with vertex setV(X)and automorphism groupA(X),we say thatXis agraphical regular representation(GRR) of a given abstract groupGif(I) G ≅ A(X) , and(II)A(X)acts onV(X) as a regular permutation group; that is, givenu, vV(X), there exists a uniqueφA(X)for whichφ(u) =v.That for any abstract groupGthere exists a graphXsatisfying (I) is well-known (cf. [3]).

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