Farey Triangle Graphs and Farey Triangle Matrices

Farey graphs as models for complex networks

Farey sequences of irreducible fractions between 0 and 1 can be related to graph constructions known as Farey graphs. These graphs were first introduced by Matula and Kornerup in 1979 and further studied by Colbourn in 1982 and they have many interesting properties: they are minimally 3-colorable, uniquely Hamiltonian, maximally outerplanar and perfect. In this paper we introduce a simple gener...

Triangle Graphs

We introduce a new class of planar graphs called triangle graphs. First, we present a formal way of constructing and characterizing triangle graphs, and then show that they characterize the adjacencies of arbitrary triangulations and they are three-colorable for a certain subclass of triangulations. Subsequently, we discuss an application of triangle graphs to the parallel nite element solution...

Farey Curves

^nr", Hn , INTRODUCTION 4. Observations and Questions References Consider the family of quadratic polynomials PA : C —> C, where A is a complex number with |A| < 1, defined by Px(z) = Xz + z. We will examine this polynomial from a dynamical point of view; that is, we will be interested in its behaviour under iteration. The first observation is that the polynomial Px has a fixed point at zero: P...

Products of Farey Fractions

A Farey Fraction Spin Chain

We introduce a new number–theoretic spin chain and explore its thermodynamics and connections with number theory. The energy of each spin configuration is defined in a translation–invariant manner in terms of the Farey fractions, and is also expressed using Pauli matrices. We prove that the free energy exists and exhibits a unique phase transition at inverse temperature β = 2. The free energy i...