The Farthest Vertices on the Suborbital Graphs via Hyperbolic Geometry

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DEĞER A. H., AKBABA Ü., Gökcan İ., TUYLU T.

17th International Geometry Symposium, Erzincan, Turkey, 19 - 22 June 2019, pp.130

• Publication Type: Conference Paper / Summary Text
• City: Erzincan
• Country: Turkey
• Page Numbers: pp.130
• Karadeniz Technical University Affiliated: Yes

Abstract

In this study, we investigate the farthest vertex where a vertex can be connected on the path of 
minimal length on the suborbital graph ????,??. The values of these special vertices are based on 
periodic continued fractions and derived by an element of the congruence subgroup of the Modular 
group Γ.
The  elements  of  Γ   sends  the  hyperbolic  lines  to  hyperbolic  lines.  So,  we  have 
represented the edges of graphs as hyperbolic geodesics in the upper half plane
H ? {?? ∈ C : ????(??) > 0},
which is the one model of hyperbolic geometry. Hyperbolic lines are as euclidean semi-circles or 
half-lines perpendicular to R as in [7].
We also give some results by using some properties of the suborbital graph ????,??  from
[1] with these special continued fractions.
Keywords: Suborbital Graphs; Modular group; Periodic Continued Fractions.