Extremal graph realizations and graph Laplacian eigenvalues

TL;DR

This paper extends the relationship between polyhedron vertices and Laplacian eigenvectors to general graphs, proposing a convex duality-based method to generate extremal graph realizations by optimizing the first non-trivial eigenvalue.

Contribution

It introduces a novel eigenvalue optimization approach for graph realization, linking spectral properties to maximal variance unit-distance configurations.

Findings

Spectral realization yields centered, unit-distance graphs with maximal variance.

The method is based on convex duality and eigenvalue optimization.

Examples demonstrate the effectiveness of the approach.

Abstract

For a regular polyhedron (or polygon) centered at the origin, the coordinates of the vertices are eigenvectors of the graph Laplacian for the skeleton of that polyhedron (or polygon) associated with the first (non-trivial) eigenvalue. In this paper, we generalize this relationship. For a given graph, we study the eigenvalue optimization problem of maximizing the first (non-trivial) eigenvalue of the graph Laplacian over non-negative edge weights. We show that the spectral realization of the graph using the eigenvectors corresponding to the solution of this problem, under certain assumptions, is a centered, unit-distance graph realization that has maximal total variance. This result gives a new method for generating unit-distance graph realizations and is based on convex duality. A drawback of this method is that the dimension of the realization is given by the multiplicity of the extremal eigenvalue, which is typically unknown prior to solving the eigenvalue optimization problem. Our results are illustrated with a number of examples.