@MISC{Roy09minimaleuclidean, author = {Aidan Roy}, title = {Minimal Euclidean representations of graphs}, year = {2009} }

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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 or 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. 1