On Single-Distance Graphs on the Rational Points in Euclidean Spaces

For positive integers n and d > 0, let denote the graph whose vertices are the set of rational points, with being adjacent if and only if the Euclidean distance between u and v is equal to d. Such a graph is deemed non-trivial if d is actually realized as a distance between points of. In this paper, we show that a space has the property that all pairs of non-trivial distance graphs and are isomorphic if and only if n is equal to 1, 2, or a multiple of 4. Along the way, we make a number of observations concerning the clique number of.