On a Graph Connecting Hyperbinary Expansions

Maurizio Brunetti, Alma D'Aniello

Le $n$ be any positive integer. A hyperbinary expansion of $n$ is a representation of $n$ as sum of powers of $2$, each power being used at most twice. 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 $n$. Such graph suggests a method to measure the non-binarity of a hyperbinary expansion. We discuss how it is related to $(p,q)$-hyperbinary expansions. Finally, we identify those integers $m\in\N$ such that the fundamental group of $A(m)$ is Abelian.