Generalized classes of suborbital graphs for the congruence subgroups of the modular group

Abstract

Let \( \Gamma \) be the modular group. We extend a nontrivial \( \Gamma \)-invariant equivalence relation on \( \widehat{\mathbb{Q}} \) to a general relation by replacing the group \( \Gamma_0(n) \) by \( \Gamma_K(n) \), and determine the suborbital graph \( \mathcal{F}^K_{u,n} \), an extended concept of the graph \( \mathcal{F}_{u,n} \). We investigate several properties of the graph, such as, connectivity, forest conditions, and the relation between circuits of the graph and elliptic elements of the group \( \Gamma_K(n) \). We also provide the discussion on suborbital graphs for conjugate subgroups of \( \Gamma \).