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


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.


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 (, 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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