Automorphism groups of graphs.

The problem of exhibiting graphs whose group is some given permutation group is exarnined, and the known answers for certain classes of groups are detailed.In the case of cyclic groups, the (negative) answer has been demonstrated by using a class of graphs here called circulants.This sarne class has also been shown to contain all graphs with transitive groups of prime degree.Here, by introducing a new class of graphs called 2-circulants, a partial characterization is made of graphs whose groups are transitive permutation groups of degree 2p for any prime p. Cayley graphs are also investigated and some aspects of this type of construction are related to the problem at hand.Included in this work is a corrected version of the published result limiting the existence of graphs with transitive abelian groups, and some additional information relevant to the cases already mentioned.Finally, a sununary of the status of the problem is presented, including a statement of some relevant theorems not here proven in detail.(iii) for Joyce Without her longsuffering and understanding it could not have been conceived, let alone written.