Minimal Euclidean representations of graphs

TL;DR

This paper investigates the minimal Euclidean dimension needed to embed a graph such that edges and non-edges are distinguished by two specific distances, providing bounds and exact formulas based on spectral properties.

Contribution

It introduces bounds and an exact formula for the Euclidean representation number of graphs using eigenvalue multiplicities and main angles.

Findings

Bound on Euclidean representation number using eigenvalue multiplicities

Exact formula for Euclidean representation number via main angles

Spectral properties determine minimal embedding dimension

Abstract

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.