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

TL;DR

This paper proves that for the integer lattice in two dimensions, non-trivial Euclidean distance graphs with different distances are never isomorphic, resolving a previously posed open problem.

Contribution

The paper provides a simple geometric construction demonstrating that no two such graphs with different distances are isomorphic in $\

Findings

Non-trivial graphs $G(\\mathbb{Z}^2, d)$ are not isomorphic for different $d$.

A geometric construction shows the non-isomorphism.

Answers a 2015 open question negatively.

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.