On the Degree Distribution of Haros Graphs

Haros graphs are 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 that is continuous and piecewise linear in subintervals defined by Farey fractions can be derived from an additional conclusion for the degree distribution of Haros graphs.

Automated summary

Haros graphs are a representation of real numbers in the unit interval using graph theory. The degree distribution of Haros graphs provides information about the topological structure and the associated real number. This article presents a comprehensive demonstration of a conjecture concerning the analytical formulation of the degree distribution. A theorem establishes the relationship between Haros graphs, the continued fraction of the associated real number, and symbolic paths in the Farey binary tree. Additionally, an expression for the degree distribution of Haros graphs can be derived from an additional conclusion that is continuous and piecewise linear in subintervals defined by Farey fractions.