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×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.

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Lovász, L, & Schrijver, A. (2017). Nullspace embeddings for outerplanar graphs. In A Journey Through Discrete Mathematics (pp. 571–591).