On conjugacy of elliptic elements and circuits in suborbitalgraphs of congruence subgroups

In this paper, we, for the first time, introduce the notion of conjugacy in suborbitalgraphs. Then we show that the conjugate elliptic elements of the modular group Γ and of congruence subgroups Γ0(p) give rise to conjugate circuits, corresponding to the relatedelliptic elements, in the Farey graph F and in the suborbital graph Fu,p of the action of Γ0(p), respectively.