Minimal Euclidean representations of graphs

A simple graph G is representable in a real vector space of dimension m if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two vertices is one of only two distinct values a or b, with distance a if the vertices are adjacent and distance b otherwise. The Euclidean representation number of G is the smallest dimension in which G is representable. In this note, we bound the Euclidean representation number of a graph to within one of its actual value using multiplicities of the eigenvalues of the adjacency matrix. We also give an exact formula for the Euclidean representation number using the main angles of the graph.