On distance graphs in rational spaces

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.