On distance graphs in rational spaces

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.