Answer to an Isomorphism Problem in ${\mathbb{Z}}^2$

Abstract

For $S\subset {\mathbb{R}}^n$ and $d>0$, denote by $G(S,d)$ the graph with vertex set $S$ with any two vertices being adjacent if and only if they are at a Euclidean distance $d$ apart. Deem such a graph to be ``non-trivial" if $d$ is actually realized as a distance between points of $S$. In a 2015 article, the author asked if there exist distinct $d_1,d_2$ such that the non-trivial graphs $G({\mathbb{Z}}^2,d_1)$ and $G({\mathbb{Z}}^2,d_2)$ are isomorphic. In our current work, we offer a straightforward geometric construction to show that a negative answer holds for this question.