# The Shortest Path Problem for the Distant Graph of the Projective Line Over the Ring of Integers

Bulletin of the Malaysian Mathematical Sciences Society, Nov 2015

The distant graph $G=G({\mathbb {P}}(Z), \vartriangle )$ of the projective line over the ring of integers is considered. The shortest path problem in this graph is solved by use of Klein’s geometric interpretation of Euclidean continued fractions. In case the minimal path is non-unique, all the possible splitting are described which allows us to give necessary and sufficient conditions for existence of a unique shortest path.

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Andrzej Matraś, Artur Siemaszko. The Shortest Path Problem for the Distant Graph of the Projective Line Over the Ring of Integers, Bulletin of the Malaysian Mathematical Sciences Society, 2015, 1-18, DOI: 10.1007/s40840-015-0273-3