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

TL;DR

This paper investigates the shortest path problem in the distant graph of the projective line over integers, utilizing Klein's geometric interpretation of continued fractions to characterize path uniqueness and solutions.

Contribution

It introduces a novel approach using Klein's geometric interpretation to solve shortest path problems and characterizes conditions for path uniqueness in this graph.

Findings

Shortest path solutions are derived using Klein's geometric interpretation.

Conditions for the existence of a unique shortest path are established.

All possible path splittings in non-unique cases are described.

Abstract

The distant graph $G = G(\mathbb{P}(Z),\triangle)$ 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.