2021/06/09 by Matt Noble, Noble, Matt
Computer Science · Mathematics · #Advanced Graph Theory Research #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Graph theory and applications #math.CO
paper · pdf · doi:10.48550/arxiv.2106.05323
arxiv created 2021/06/09 · openalex publication_date 2021/06/09 · arxiv updated 2021/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For S ⊂ ℝn 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 d1, d2 such that the non-trivial graphs G(ℤ2, d1) and G(ℤ2, d2) are isomorphic. In our current work, we offer a straightforward geometric construction to show that a negative answer holds for this question.