A note on the Ádám conjecture for double loops

We study the condition for isomorphism between double loops (or chordal rings) which is a simple kind of circulant graph. We show that the family of n nodes double loops having chord length below min{n/4, φ(n)/2} has the Ádám property, answering partially a long-standing conjecture. We also give a simpler proof of the fact that all circulant graphs with a prime number of nodes have the Ádám property.