In this paper, we prove several results on the Cayley isomorphism problem concerning undirected graphs. Let ϕ be Euler's phi function. The main result is the following theorem. Let q1,…,qs be distinct primes and p a prime such that p2<q1 and p2q1…qi<qi+1 for all 1⩽i⩽s−1. Let n=q1…qs. If gcd(np,ϕ(np))=1, then is a CI-group with respect to graphs.