Isomorphisms of connected Cayley digraphs

Let G be a finite group and let Cay(G, S) be a Cayley graph of G. The graph Cay(G, S) is called a CI-graph of G if, for any T subset of G, S = T-alpha for some alpha epsilon Aut(G) only when Cay(G, S) congruent to Cay(G, T). In this paper, we study the isomorphism problem of connected Cayley graphs: to determine the groups G (or the types of Cayley graphs for a given group G) for which all connected Cayley graphs for G are CI-graphs.