Nullspace embeddings for outerplanar graphs

TL;DR

This paper explores the relationship between geometric embeddings of outerplanar graphs and matrix spectra, providing an algorithmic approach to characterize such embeddings via special matrices.

Contribution

It introduces a new class of matrices called 'good' G-matrices and establishes a method to determine outerplanar embeddings through their nullspace representations.

Findings

Nullspace representations can produce outerplanar embeddings for 2-connected graphs.

Existence of a 'good' G-matrix with corank 2 indicates an outerplanar embedding.

If not, a 'good' G-matrix with corank 3 exists, characterizing the graph's embedding properties.

Abstract

We study relations between geometric embeddings of graphs and the spectrum of associated matrices, focusing on outerplanar embeddings of graphs. For a simple connected graph $G=(V,E)$, we define a "good" $G$-matrix as a $V\times V$ matrix with negative entries corresponding to adjacent nodes, zero entries corresponding to distinct nonadjacent nodes, and exactly one negative eigenvalue. We give an algorithmic proof of the fact that it $G$ is a 2-connected graph, then either the nullspace representation defined by any "good" $G$-matrix with corank 2 is an outerplanar embedding of $G$, or else there exists a "good" $G$-matrix with corank 3.