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

Abstract

© Journal “Algebra and Discrete Mathematics” Let Γ be the modular group. We extend a nontrivial Γ-invariant equivalence relation on ℚ to a general relation by replacing the group Γ 0 (n) by Γ K (n), and determine the suborbital graph F Ku,n , an extended concept of the graph 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 Γ K (n). We also provide the discussion on suborbital graphs for conjugate subgroups of Γ.