Minimal Euclidean representations of graphs

A simple graph GG is representable   in a real vector space of dimension mm, if there is an embedding of the vertex set in the vector space such that the Euclidean distance between any two distinct vertices is one of only two distinct values, αα and ββ, with distance αα if the vertices are adjacent and distance ββ otherwise. The Euclidean representation number   of GG is the smallest dimension in which GG is representable. In this note, we bound the Euclidean representation number of a graph 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.