Minimal Euclidean representations of graphs

Details Minimal Euclidean representations of graphs Minimal Euclidean representations of graphs

2008-12-19

arxiv.org/abs/0812.3707v3

Mathematics

Combinatorics

15 pages, one figure; errors corrected from previous version

Scientific paper

A simple graph G is said to be 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 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 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.

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