On the action of non-Abelian groups on graphs

Abstract Finite groups may be classified as to whether or not they have regular representations as automorphism groups of graphs. Such a classification is begun here for non-Abelian groups (classification of Abelian groups being already in the literature) with a classification of the dihedral groups, the groups of order p 3 for p an odd prime (except for one group of order 27), and the generalized dicyclic groups. It is further shown that the class of groups (other than C 2 ) having the above regular representations is closed under the operation of direct product.