Geodesic
Generalized classes of suborbital graphs for the congruence subgroups of the modular group
Algebra and discrete mathematics, Tome 27 (2019) no. 1, pp. 20-36

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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 $\Gamma$.
Keywords: modular group, congruence subgroups, suborbital graphs. 
@article{ADM_2019_27_1_a3,
     author = {Pradthana Jaipong and Wanchai Tapanyo},
     title = {Generalized classes of suborbital graphs for the congruence subgroups of the modular group},
     journal = {Algebra and discrete mathematics},
     pages = {20--36},
     publisher = {mathdoc},
     volume = {27},
     number = {1},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ADM_2019_27_1_a3/}
}
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Pradthana Jaipong; Wanchai Tapanyo. Generalized classes of suborbital graphs for the congruence subgroups of the modular group. Algebra and discrete mathematics, Tome 27 (2019) no. 1, pp. 20-36. http://geodesic.mathdoc.fr/item/ADM_2019_27_1_a3/