×

The shortest path problem for the distant graph of the projective line over the ring of integers. (English) Zbl 1385.51003

The projective line over a ring gives rise to a distant graph \(G=(\mathbb{P}(R),\Delta)\) with the points of \(\mathbb{P}(R)\) as vertices and the undirected pairs of distant points as edges. The authors solve the shortest path problem for this graph in case \(R\) is the ring \(\mathbb{Z}\) of integers, using a geometric interpretation of continued fractions. They also formulate necessary and sufficient conditions for the existence of a unique shortest path.

MSC:

51C05 Ring geometry (Hjelmslev, Barbilian, etc.)
05C12 Distance in graphs
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Blunck, A., Havlicek, H.: Projective representations I: projective lines over a ring. Abh. Math. Semin. Univ. Hambg. 70, 287-299 (2000) · Zbl 0992.51001 · doi:10.1007/BF02940921
[2] Blunck, A., Havlicek, H.: The connected components of the projective line over a ring. Adv. Geom. 1, 107-117 (2001) · Zbl 0992.51003 · doi:10.1515/advg.2001.008
[3] Blunck, A., Havlicek, H.: On distant-isomorphisms of projective lines. Aequationes Math. 69, 146-163 (2001) · Zbl 1073.51004 · doi:10.1007/s00010-004-2745-7
[4] Blunck, A., Herzer, A.: Kettengeometrien-Eine Einführung. Shaker Verlag, Aachen (2005) · Zbl 1095.51001
[5] Cohn, P.H.: On the structure of the \[GL_2\] GL2 of a ring. Inst. Hautes Etudes Sci. Publ. Math. 30, 5-53 (1966) · Zbl 0144.26301 · doi:10.1007/BF02684355
[6] Herzer, A.; Buekenhout, F. (ed.), Chain geometries, 781-842 (1995), Elsevier · Zbl 0829.51003 · doi:10.1016/B978-044488355-1/50016-5
[7] Popescu-Pampu, P.; Brasselet, J-P (ed.); Suwa, T. (ed.), The geometry of continued fractions and the topology of surface singularities, No. 46, 119-195 (2007), Tokyo · Zbl 1129.14046
[8] Siemaszko, A., Wojtkowski, M.P.: Counting Berg partitions. Nonlinearity 24, 2383-2403 (2011) · Zbl 1237.37038 · doi:10.1088/0951-7715/24/9/001
[9] Siemaszko, A., Wojtkowski, M.P.: Counting Berg partitions via Sturmian words and substitution tilings. European Congress of Mathematics Kraków, pp. 779-790. European Mathematical Society Publishing House (2014), 2-7 July 2012 · Zbl 1364.37045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.