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On Single-Distance Graphs on the Rational Points in Euclidean Spaces

Part of: Graph theory Elementary number theory

Published online by Cambridge University Press:  29 July 2020

Sheng Bau ,
Peter Johnson  and
Matt Noble
Sheng Bau
Affiliation:
School of Mathematics, Statistics and Computer Science, University of Kwazulu-Natal, Durban, South Africa e-mail: baus@ukzn.ac.za
Peter Johnson
Affiliation:
Department of Mathematics and Statistics, Auburn University, Auburn, Alabama e-mail: johnspd@auburn.edu
Matt Noble*
Affiliation:
Department of Mathematics and Statistics, Middle Georgia State University, Macon, Georgia
*
e-mail: matthew.noble@mga.edu
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Abstract

For positive integers n and d > 0, let $G(\mathbb {Q}^n,\; d)$ denote the graph whose vertices are the set of rational points $\mathbb {Q}^n$, with $u,v \in \mathbb {Q}^n$ being adjacent if and only if the Euclidean distance between u and v is equal to d. Such a graph is deemed “non-trivial” if d is actually realized as a distance between points of $\mathbb {Q}^n$. In this paper, we show that a space $\mathbb {Q}^n$ has the property that all pairs of non-trivial distance graphs $G(\mathbb {Q}^n,\; d_1)$ and $G(\mathbb {Q}^n,\; d_2)$ are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of $G(\mathbb {Q}^n,\; d)$.

Keywords

Euclidean distance graphrational pointsgraph isomorphismclique numberregular simplex

MSC classification

Primary: 05C62: Graph representations (geometric and intersection representations, etc.)
Secondary: 11A15: Power residues, reciprocity
Type
Article
Information
Canadian Mathematical Bulletin , Volume 64 , Issue 1 , March 2021 , pp. 13 - 24
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Copyright
© Canadian Mathematical Society 2020

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