Minimal Euclidean representations of graphs

Abstract

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 distinct vertices is one of only two distinct values, α and β , with distance α if the vertices are adjacent and distance β 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 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.