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

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 $\Gamma$.