Isomorphisms of graphs of Hyperbinary Expansions and Efficient Algorithms for Stern's Diatomic Sequence

TL;DR

This paper explores the structure of graphs representing hyperbinary expansions, establishes their isomorphism properties, and derives formulas and algorithms related to Stern's diatomic sequence.

Contribution

It provides a new simple description of the graph structure and proves that isomorphic graphs correspond only to identical integers, also deriving related formulas and discussing algorithms.

Findings

Graphs $A(n)$ have a simple structural description.

Isomorphism of $A(n)$ graphs implies $m=n$ for even $m,n$.

Derived formulas connect to Stern's diatomic sequence.

Abstract

To investigate hyperbinary expansions of a nonnegative integer~$n$, an edge-labeled directed graph $A(n)$ has recently been introduced. After pointing out some new simple facts about its cyclomatic number, we give a relatively simple description of its structure and prove that if $m,n$ are even numbers for which $A(n)$ and $A(m)$ are isomorphic as edge-labeled graphs, then $m=n$. From the structure of $A(n)$ we also derive a formula related to Stern's diatomic sequence, and in the same vein discuss some algorithms that recently appeared in the literature.