On the degree distribution of Haros graphs

TL;DR

This paper investigates the degree distribution of Haros graphs, revealing its connection to continued fractions and Farey trees, and provides an analytical, piece-wise linear formula for it.

Contribution

It proves a conjecture linking Haros graphs' degree distribution to continued fractions and Farey trees, offering an explicit analytical expression.

Findings

Established the relationship between Haros graphs and continued fractions.

Derived a continuous, piece-wise linear formula for the degree distribution.

Validated the conjecture through rigorous mathematical demonstration.

Abstract

Haros graphs is a graph-theoretical representation of real numbers in the unit interval. The degree distribution of the Haros graphs provides information regarding the topological structure and the associated real number. This article provides a comprehensive demonstration of a conjecture concerning the analytical formulation of the degree distribution. Specifically, a theorem outlines the relationship between Haros graphs, the corresponding continued fraction of its associated real number, and the subsequent symbolic paths in the Farey binary Tree. Moreover, an expression continuous and piece-wise linear in subintervals defined by Farey fractions can be derived from an additional conclusion for the degree distribution of Haros graphs.