On a graph connecting hyperbinary expansions

Let 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 2 N such that the fundamental group of A(m) is Abelian.