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

Abstract

Let Γ be the modular group. We extend a non-trivial Γ-invariant equivalence relation on Qb to a general relation by replacing the group Γ0(n) by ΓK(n), and determine the suborbital graph F K u,n, an extended concept of the graph Fu,n. We investigate several properties of the graph, such as, connectivity, forest condi- tions, and the relation between circuits of the graph and elliptic elements of the group ΓK(n). We also provide the discussion on suborbital graphs for conjugate subgroups of Γ.