Abelian permutation groups with graphical representations

In this paper we characterize those automorphism groups of colored graphs and digraphs that are abelian as abstract groups. This is done in terms of basic permutation group properties. Using Schur’s classical terminology, what we provide is characterizations of the classes of 2-closed and 2^* -closed abelian permutation groups. This is the first characterization concerning these classes since they were defined.