On Suborbital Graphs for Some Subgroups of the Normalizer of $$\Gamma _0(qp^2)$$

Abstract

In this paper,we find some certain relationships between the lengths of the circuits in suborbital graphs $$F_{u,N}$$ for $$\Lambda (N)$$ on $$\hat{\mathbf {Q}}(N)$$ and the periods of the group $$\Lambda (N)$$ , generated by $$\Gamma _0(N)$$ and an Atkin-Lehner involution, which is a subgroup of the normalizer $$N(\Gamma _0(N))$$ of $$\Gamma _0(N)$$ in $$PSL(2,\mathbf {R})$$ where N will be of the form $$qp^2$$ , where $$q=1$$ or prime and p is prime $$>3$$ and $$p\ne q.$$ In this case the action of the nomalizer on the extended rational numbers $$\hat{\mathbf {Q}}$$ is not transitive. And finally, we give some number theoretical results.