On distance graphs in rational spaces

TL;DR

This paper investigates rational distance graphs defined by positive definite quadratic forms, determining their clique numbers and proving a rational analogue of the Beckman–Quarles theorem, which characterizes unit-preserving maps as isometries.

Contribution

It introduces methods to compute clique numbers of rational distance graphs and establishes that unit-preserving maps on rational spaces are isometries, extending classical geometric results.

Findings

Exact clique numbers for rational distance graphs are derived.

Rational analogue of Beckman–Quarles theorem proven.

Unit-preserving maps on rational spaces are isometries.

Abstract

For any positive definite rational quadratic form $q$ of $n$ variables let $G(\mathbb{Q}^n, q)$ denote the graph with vertices $\mathbb{Q}^n$ and $x, y \in \mathbb{Q}^n$ connected iff $q(x - y) = 1$. This notion generalises standard Euclidean distance graphs. In this article we study these graphs and show how to find the exact value of clique number of the $G(\mathbb{Q}^n, q)$. We also prove rational analogue of the Beckman--Quarles theorem that any unit-preserving mapping of $\mathbb{Q}^n$ is an isometry.