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Results in Mathematics

, 73:141 | Cite as

The Cayley Property of Some Distant Graphs and Relationship with the Stern–Brocot Tree

  • Andrzej Matraś
  • Artur SiemaszkoEmail author
Open Access
Article
  • 101 Downloads

Abstract

One of the graphs associated with any ring R is its distant graph \(G(R,\Delta )\) with points of the projective line \(\mathbb {P}(R)\) over R as vertices. We prove that the distant graph of any commutative, Artinian ring is a Cayley graph. The main result is the fact that \(G(\mathbb Z,\Delta )\) is a Cayley graph of a non-artinian commutative ring. We indicate two non-isomorphic subgroups of \(PSL_2(\mathbb Z)\) corresponding to this graph.

Keywords

Cayley graphs distant graphs projective lines over rings Stern–Brocot tree 

Mathematics Subject Classification

Primary 05C25 Secondary 51C05 

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Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Mathematics and Computer ScienceUniversity of Warmia and Mazury in OlsztynOlsztynPoland

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