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

Abstract

The distant graph G = G(P(Z ), ) 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.

The Cayley Graph of Neumann Subgroups

“…This paper is a continuation of our project to find all Cayley representations of Γ Z in P GL(2, Z) ( [10], [11], [12]) and it completes the series of works devoted to the description of Cayley's groups of P(Z). Because automorphisms groups of Γ Z is P GL(2, (Z) it gives all Cayley groups of Γ Z .…”

The Cayley Graph of Neumann Subgroups

“…After sending our paper we found works of Magnus [9], Brenner, Lyndon [5,6] and realise that our research overlap in part with those in these works. This paper is a continuation of our project to find all Cayley representations of Γ Z in P GL(2, Z) [10][11][12] and it completes the series of works devoted to the description of Cayley's groups of P(Z). Because automorphisms groups of Γ Z is P GL(2, Z) it gives all Cayley groups of Γ Z .…”

The Distant Graph of the Ring of Integers and Its Representations in the Modular Group

“…It is also called the distant graph of the ring of integers. The properties of distant graphs have been studied in many papers, for instance [1][2][3]5,6]. According to the general definition of the distant graph (see i.e.…”