Infinite Paths of Minimal Length on Suborbital Graphs for Some Fuchsian Groups

Khuanchanok Chaichana,Pradthana Jaipong

doi:10.1155/2019/7652729

Khuanchanok Chaichana, Pradthana Jaipong

Open Access

https://doi.org/10.1155/2019/7652729

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Abstract

In this study, we work on the Fuchsian group Hm where m is a prime number acting on mℚ^ transitively. We give necessary and sufficient conditions for two vertices to be adjacent in suborbital graphs induced by these groups. Moreover, we investigate infinite paths of minimal length in graphs and give the recursive representation of continued fraction of such vertex.

Highlights

Introduction eHecke group, H(λ), introduced by Hecke in [1], is the group generated by the two Mobius transformationsR(z) − 1, z (1) √ az + b mT(z) √, c mz + d a, b, c, d ∈ Z, ad − bcm 1, (2) a mz + b T(z)√, cz + d m a, b, c, d ∈ Z, adm − bc 1. (3)Rosen [3] showed that the above√t wo√ transformations need not to be in H(λ) if λ ≠ 1, 2, 3
We study on the Fuchsian group H( m ) where m is a prime number
We start investigating vertices in the infinite path of minimal length by determining√th e farthest vertex which can be joined with the vertex (u/v) m

Summary

Introduction

We investigate infinite paths of minimal length in graphs and give the recursive representation of continued fraction of such vertex. We focus on the infinite path√o f minimal length in the suborbital graph G(∞, (u/v) m ) where (m, v) 1.

Results

Conclusion