On Suborbital Graphs for Some Subgroups of the Normalizer of Γ0(qp²)

In this paper,we find some certain relationships between the lengths of the circuits in suborbital graphs Fu,N for Λ (N) on Q̂(N) and the periods of the group Λ (N), generated by Γ0(N) and an Atkin-Lehner involution, which is a subgroup of the normalizer N(Γ0(N)) of Γ0(N) in PSL(2,R) where N will be of the form qp², where q = 1 or prime and p is prime 3 and p ≠ q. In this case the action of the nomalizer on the extended rational numbers Q̂ is not transitive. And finally, we give some number theoretical results.