Graphs on which dihedral, quaternion, and abelian groups act vertex and/or edge transitively and applications to tensor product

The graphs on which dihedral, quaternion, and abelian groups act vertex and/or edge transitivity are completely characterized. The vertex transitive graphs belong to one of three families--the well known circulant graphs, the metacirculant graphs constructured by Alspach and Parsons, and a family constructed using a generalization of Alspach's and Parsons' construction. If one of the selected groups acts both vertex and edge transitively on a graph, then it is shown that the graph is the disjoint union of some number of copies of a given cycle. The graphs on which one of the selected groups acts edge transitively but not vertex transitively fall into two broad--disjoint copies of a complete bipartite graph and disjoint copies of a "pseudo-cycle," a graph which is related to the tensor product of a complete bipartite graph and an even cycle.;The full automorphism group of a pseudo-cycle is then found. It is shown that this group depends upon several parameters used in creating the pseudo-cycle, and in the most general case, the automorphism group is not quite what one might expect it to be. Next, these results are used to find the full automorphism group of the tensor product of any cycle with a complete bipartite graph. Finally, it is shown that the technique used to find the full automorphism group of a pseudo-cycle can be used to find the automorphism groups of certain tensor products. In particular, conditions on a graph $Gamma$ are found that insure that the full automorphism groups of the tensor products of $Gamma$ with the complete graph $Ksb{n}$, the complete graph with loops $Ksbsp{n}{*}$, and the complete bipartite graph $Ksb{n,n}$ are as small as possible.