Nullspace embeddings for outerplanar graphs
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×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.