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\begin{Abstract}
	Let $ \Gamma $ be the group of modular. Then $ \Gamma $ acts transitively on $ \widehat{\mathbb{Q}} $ by linear fractional transformation. We generalize the nontrivial $ \Gamma $-invariant equivalence relation on $ \widehat{\mathbb{Q}} $ related to $ \Gamma_0(n) $ by replacing the subgroup $ \Gamma_0(n) $ of $ \Gamma $ by the subgroup $ \Gamma_K(n) $. Then we determine the subgraph $ \mathcal{F}^K_{u,n} $ of the suborbital graph $ \mathcal{G}_{u,n} $ for $ \Gamma $. It is the subgraph of  $ \mathcal{G}_{u,n} $ restricted on the block $ [\infty]_K $, the equivalence class in $ \widehat{\mathbb{Q}} $ containing $ \infty $. This is an extended concept of the graph $ \mathcal{F}_{u,n} $. We investigate several properties of the graph, such as, connectivity, forest conditions, graph coloring, 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 $.
	
	In the case of the extended modular group $ \widehat{\Gamma} $, we replace the subgroup $ \widehat{\Gamma}_0(n) $ by $ \widehat{\Gamma}_K(n) $. By the similar way, we obtain the subgraph $ \widehat{\mathcal{F}}^K_{u,n} $ of the suborbital graph $ \widehat{\mathcal{G}}_{u,n} $ for $ \widehat{\Gamma} $. This is a generalization of the graph $ \widehat{\mathcal{F}}_{u,n} $. We provide the relations between the graphs $ {\mathcal{F}}^K_{u,n} $ and $ \widehat{\mathcal{F}}^K_{u,n} $. Then many properties of $ {\mathcal{F}}^K_{u,n} $ and $ \widehat{\mathcal{F}}_{u,n} $ are extended to $ \widehat{\mathcal{F}}^K_{u,n} $.
	
	In the part of buttonings of graphs, we examine the lengths of maximal buttonings of certain finite undirected connected graphs including the special subgraphs of the Farey graph $ \mathcal{F}=\mathcal{G}_{1,1} $ and the Stern-Brocot tree which advantages in some parts of suborbital graphs.
\end{Abstract}