For any d > 0, define $G({\mathbb{Q}}^n,d)$ to be the graph whose vertices are points of the rational space ${\mathbb{Q}}^n$ with any two vertices being adjacent if and only if they are a Euclidean distance $d$ apart. Such a graph is only of interest if $d$ is a distance actually realized between points of ${\mathbb{Q}}^n$, so we might as well assume that is the case. In this talk, we will ask for which $n$ and distances $d_1,d_2$ the graphs $G({\mathbb{Q}}^n,d_1)$ and $G({\mathbb{Q}}^n,d_2)$ are isomorphic. A resolution will be given for $n\le 4$, and we will then present, by way of drawing a bunch of pictures, a method that, perhaps with some ingenuity, could be extended to answer this question for general $n$.

Keywords: Euclidean distance graph, rational points, isometry, graph isomorphism