Investigation Of NDCI-property Of Cyclic Groups

The symmetry of a graph has an important research position in graph theory and it mainly investigates its symmetry by the automorphism groups of a graph.In the study of the symmetry of graphs,Cayley graph is a representative.Let G be a finite group and let S be a nonempty subset of G not containing the identity element 1.The Cayley digraph Cay(G,S)of G with respect to S is defined to have vertex set G,and directed edge set {(g,sg)|g ∈ G,s ∈ S}.If S=S-1,Cay(G,S)can be viewed as undirected graph by identifying two opposite directed edges as a undirected edge.The isomorphism problem of Cayley graphs,that is CI-problem,is an important branch of Cayley graphs theory.A Cayley digraph Cay(G,S)is called a CI-graph of G if,for any Cayley digraph Cay(G,T),whenever Cay(G,S)≌ Cay(G,T)we have Sσ=T for some σ ∈ Aut(G).A Cayley digraph Cay(G,S)is called normal if the right regular representation R(G)of G is a normal subgroup of Aut(Cay(G,S)).If all Cayley digraphs of G are CI-graph,then G is called a DCI-group,and if all normal Cayley digraphs of G are CI-graphs,then G is called NDCI-group.In 1967,Adam proposed the well-known conjecture:all cyclic groups are DC I-groups.Under the efforts of many experts over 30 years,in 1997 Muzychuk classified all cyclic DCI-groups.In this paper,we mainly investigate cyclic NDCIgroups,and prove that cyclic groups Z2,Z4 and cyclic groups of odd order are NDCI-groups,but Z2n(n≥3)are not NDCI-groups.