Asymptotic automorphism groups of Cayley digraphs and graphs of abelian groups of prime-power order

We show that almost every Cayley graph ▫$Gamma$▫ of an abelian group ▫$G$▫ of odd prime-power order has automorphism group as small as possible. Additionally, we show that almost every Cayley (di)graph ▫$Gamma$▫ of an abelian group ▫$G$▫ of odd prime-power order that does not have automorphism group as small as possible is a normal Cayley (di)graph of ▫$G$▫ (that is, ▫$G_L triangleleft {rm Aut}(Gamma))$▫

On the Automorphism Groups of Almost All Circulant Graphs and Digraphs

We attempt to determine the structure of the automorphism group of a generic circulant graph. We first show that almost all circulant graphs have automorphism groups as small as possible. Dobson has conjectured that almost all of the remaining circulant (di)graphs (those whose automorphism groups are not as small as possible) are normal circulant (di)graphs. We show this conjecture is not true in general, but is true if we consider only those circulant (di)graphs whose orders are in a “large” subset of integers. We note that all non-normal circulant (di)graphs can be classified into two natural classes (generalized wreath products, and deleted wreath type), and show that neither of these classes contains almost every non-normal circulant digraph