TY - JOUR T1 - Trees of the Normalizer of Modular Group in the Picard Group AU - Yazıcı Gözütok, Nazlı AU - Zengin, İlgıt AU - Güler, Bahadır Özgür PY - 2018 DA - December JF - Turkish Journal of Mathematics and Computer Science JO - TJMCS PB - Matematikçiler Derneği WT - DergiPark SN - 2148-1830 SP - 63 EP - 70 VL - 9 LA - en AB - In this study, we investigate trees arising from the imprimitive action of the normalizer of Modulargroup in the Picard group on extended rational numbers. We determine the farthest vertex from a given vertexin hyperbolic paths of minimal lengths. We also include some results of the suborbital graph F_{u,N} related to acontinued fraction representation of a rational number. KW - Normalizer of the modular group KW - suborbital graphs KW - continued fraction CR - Akbaş, M., On Suborbital Graphs for the Modular Group, Bulletin of the London Mathematical Society 33(6)(2001), 647–652. CR - Akbaş, M., Bas¸kan, T., Suborbital graphs for the normalizer of 􀀀0(N), Turk J Math, 20(1996), 379–387. CR - Beşenk, M., The action of S L(2;C) on hyperbolic 3-space and orbital graphs, Graphs Combin., 34(4)(2018), 545–554. CR - Bigg, N.L., White, A.T., Permutation groups and combinatorial structures, London Mathematical Society Lecture Note Series, 33, CUP, Cambridge, 1979. CR - Chaichana K, Jaipong P, Suborbital Graphs for Congruence Subgroups of the Extended Modular Group and Continued Fractions, Proceedings of AMM, 20(2015), 86–95. CR - Cuyt A. et al., Handbook of Continued Fractions for Special Functions, Springer, New York, 2008. CR - Değer AH, Beşenk M, Güler BO, On suborbital graphs and related continued fractions, Appl. Math. Comput., 218(3)(2011), 746–750. CR - Değer AH, Vertices of paths of minimal lengths on suborbital graphs, Filomat, 31(4)(2017), 913–923. CR - Jones GA, Singerman D, Complex functions: an algebraic and geometric viewpoint, Cambridge University Press, Cambridge, 1987. CR - Jones GA, Singerman D, Wicks K, The modular group and generalized Farey graphs. London Math. Soc. Lecture Note Series 160(1991), 316–338. CR - Güler, B.Ö . et al., Elliptic elements and circuits in suborbital graphs, Hacet. J. Math. Stat., 40(2)(2011), 203–210. CR - Güler, B.Ö ., Kör, T., Şanlı, Z.: Solution to some congruence equations via suborbital graphs. Springerplus, 2016(5)(2016), 1-11. CR - Kader, S., Circuits in suborbital graphs for the normalizer. Graphs Combin. 33(6)(2017), 1531–1542. CR - Keskin, R., Suborbital graphs for the normalizer 􀀀0(m). European Journal of Combinatorics 27(2)(2006), 193–206. CR - Keskin, R., Demirt¨urk, B., On suborbital graphs for the normalizer of 􀀀0(N). The Electronic Journal of Combinatorics 16(1)(2009), 1–18. CR - Köroğlu, T., Güler, B.Ö., Şanlı, Z., Suborbital graphs for the Atkin-Lehner group. Turk J Math. 41(2017), 235–243. CR - Köroğlu, T., Güler, B.Ö ., Şanlı, Z., Some Generalized Suborbital Graphs. Turk. J. Math. Comput. Sci., 7(2017), 90–95. CR - Kushwaha, S.; Sarma, R.; Continued fractions arising from F1;3. Ramanujan J. 46(3)(2018), 605–631. CR - Nathanson, M.B., A forest of linear fractional transformations. Int. J. Number Theory 11(4)(2015), 1275–1299. CR - Ponton, L., Two trees enumerating the positive rationals. Integers 18A(2018), Paper No. A17, 16 pp. CR - Sarma R, Kushwaha S, Krishnan R, Continued fractions arising from F1;2. J. Number Theory 154(2015), 179–200. CR - Wall H.S., Analytic Theory of Continued Fractions, first ed., D.Van Nostrand Co, New York, 1948. CR - Yazıcı Gözütok, N., Güler, B.Ö ., Suborbital Graphs of the Normalizer of Modular Group in the Picard Group, Iran J Sci Technol Trans Sci 42(4)(2018), 2167–2174. UR - https://dergipark.org.tr/en/pub/tjmcs/article/453611 L1 - https://dergipark.org.tr/en/download/article-file/607668 ER -