In this paper, we examine some properties of suborbital graphs for the Picard group. We obtain edge and circuit conditions, then propose a conjecture for the graph to be forest. This paper is an extension of some results in (Jones et al. in The Modular Group and Generalized Farey Graphs, pp. 316-338, 1991).

In this study, we investigate suborbital graphs G_u,n of the normalizer Γ_B (N) of Γ₀ (N) in PSL(2, ℝ) for N = 2^α3^β where α = 1, 3, 5, 7, and β = 0 or 2. In these cases the normalizer becomes a triangle group and graphs arising from the action of the normalizer contain quadrilateral circuits. In order to obtain graphs, we first define an imprimitive action of Γ_B (N) on $\stackrel{\wedge }{\mathbb{Q}}$ using the group ${\text{Г}}_{{\rm B}}^{+}$ (N) and then obtain some properties of the graphs arising from this action.