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 if and only if $q(x-y)=1$. 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 bijection of ${\mathbb{Q}}^n$ onto itself is an isometry.

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