Farey Graphs as Models for Complex Networks

TL;DR

This paper introduces a new method to generate Farey graphs and analyzes their topological properties, demonstrating their suitability as models for complex networks.

Contribution

It presents a simple generation method for Farey graphs and provides an analytical study of their topological characteristics.

Findings

Farey graphs are minimally 3-colorable and uniquely Hamiltonian.

They are maximally outerplanar and perfect.

The graphs exhibit properties suitable for modeling complex networks.

Abstract

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 generation method for a Farey graph family, and we study analytically relevant topological properties: order, size, degree distribution and correlation, clustering, transitivity, diameter and average distance. We show that the graphs are a good model for networks associated with some complex systems.