  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.
