Abstract: Let G be a finite group, H a core-free subgroup of G and D a union of several double-cosets of the form HgH with g$\notin$H such that D=D^-1. Let Cos(G, H, D) be the coset graph of G with respect to H and D, and let A=Aut(Cos(G, H, D)). Denote the right multiplication action of G on Ω=[G:H] by R_H (G), the set of right cosets of H in G, and denote the automorphism of G induced by the conjugate of g∈G on G by σ(g). In this paper, it is shown that N_A(R_H(G))=R_H(G)Aut(G, H, D) and R_H (G)∩ Aut(G, H, D)=I(H), where Aut(G, H, D)= {α∈Aut(G)|H^α=H, D^α=D}and I(H)={σ(h)|h∈H}, and it is also shown that Cos(G, H, D) is a CI-graph if and only if for any σ∈S_Ω with R_H(G)^σ≤A, and that there exists a∈A such that R_H(G)^a=R_H(G)^σ. The CI-propety problem and isomorphism count problem of a class coset graphs on linear groups are considered in application.

Key words: rc-transitive graph, coset graph, Cayley graph