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Suborbital graphs of the congruence subgroup \(\varGamma ^{0}(N)\). (English) Zbl 1409.05101

Authors’ abstract: In this paper, for any positive integer \(N\), we determine the edge conditions of suborbital graphs for the congruence subgroup \(\Gamma^{0}(N)\), \( G_{p,q}\), and the suborbital graph of the block \([\infty]\), \(F_{p,q}\), which is a subgraph of \(G_{p,q}\). Then we give the necessary and sufficient conditions for graphs to contain a triangle, which eventually leads to the conditions for it to be a forest.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
11A07 Congruences; primitive roots; residue systems
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References:

[1] Akbas, M.: On suborbital graphs for the modular group. Bull. Lond. Math. Soc. 33, 647-652 (2001) · Zbl 1023.05073 · doi:10.1112/S0024609301008311
[2] Akbas, M., Singerman, D.: The Signature of the normalizer of \[\varGamma_0(N)\] Γ0(N). Bull. Lond. Math. Soc. Lecture Note Series 165, 77-86 (1992) · Zbl 0813.20055
[3] Biggs, N.L., White, A.T.: Permutation groups and combinatorial structures. Lond. Math. Soc. Lecture Notes 33, Cambridge University Press, Cambridge (1979) · Zbl 0415.05002
[4] Güler, B.Ö., Kader, S.: On the action of \[\varGamma^0(N)\] Γ0(N) on \[\hat{\mathbb{Q}}\] Q^. Note Mat. 30, 141-148 (2010)
[5] Guler, Bahadir O., Kader, Serkan, Besenk, Murat: On suborbital graphs of the congruence subgroup \[\varGamma_0(N)\] Γ0(N). Int. J. Comput. Math. Sci. 2(3), 153-156 (2008)
[6] Jones, G.A., Singerman, D., Wicks, K.: The modular group and generalized Farey graphs. Lond. Math. Soc. Lecture note Ser. 160, 316-338 (1991) · Zbl 0728.20040
[7] Sims, C.C.: Graphs and finite permutation groups. Math. Z. 95, 76-86 (1967) · Zbl 0244.20001 · doi:10.1007/BF01117534
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