On a Graph Connecting Hyperbinary Expansions

TL;DR

This paper investigates the properties of a graph constructed from hyperbinary representations of integers, focusing on the conditions under which its fundamental group is abelian, revealing structural insights into these representations.

Contribution

It introduces a novel graph model for hyperbinary expansions and characterizes integers with abelian fundamental groups in this context.

Findings

Identifies integers with abelian fundamental groups in the hyperbinary expansion graph

Defines a new graph structure connecting hyperbinary representations

Provides structural properties of the graph A(n)

Abstract

Le n be any positive integer. A hyperbinary expansion of n is are presentation of n as sum of powers of 2, each power being used at most twice. In this paper we study some properties of a suitable edge-coloured and vertex-weighted oriented graph A(n) whose nodes are precisely the several hyperbinary representations of n. In particular, we identify those integers m in N such that the fundamental group of A(m) is abelian.