Affiliation:
1. ARTVİN ÇORUH ÜNİVERSİTESİ, FEN-EDEBİYAT FAKÜLTESİ
2. KARADENİZ TEKNİK ÜNİVERSİTESİ, FEN FAKÜLTESİ
Abstract
In this study, suborbital graphs, $G_{u,N}$ and $F_{u,N}$ are examined. Modular group $\Gamma$ and its act on $\widehat{\mathbb{Q}}$ are studied. Lorentz matrix that gives the vertices obtained under the classical matrix multiplication in the suborbital graph $F_{u,N}$ is analysed with the Lorentz matrix multiplication. Lorentz matrix written as Möbius transform is normalized and the type of the transform is researched. Moreover, a different element of Modular group $\Gamma$ is scrutinized. The vertices on the path starting with $\infty$ are obtained under this element and the Lorentz matrix multiplication. For this path, it is shown that the vertices obtained in $F_{u,N}$ under the Lorentz matrix multiplication with the Lorentz matrix satisfied the farthest vertex condition for the previous vertex.
Reference14 articles.
1. C. C. Sims, Graphs and Finite Permutation Groups, Mathematische Zeitschrift 95 (1967) 76–86.
2. P. M. Neumann, Finite Permutation Groups, Edge Coloured Graphs and Matrices, Topics in Group Theory and Computation, Academic Press, London, 1977.
3. T. Tsukuzu, Finite Groups and Finite Geometries, Cambridge University Press, Cambridge, 1982.
4. G. A. Jones, D. Singerman, K. Wicks, The Modular Group and Generalized Farey Graphs, London Mathematical Society Lecture Note Series 160 (1991) 316–338.
5. M. Akbaş, On Suborbital Graphs for Modular Group, Bulletin of the London Mathematical Society 33 (2001) 647-652.