The Strong Arnold Property for 4-connected Flat Graphs

We show that if G = (V,E) is a 4-connected flat graph, then any real symmetric V × V matrix M with exactly one negative eigenvalue and satisfying, for any two distinct vertices i and j, Mij < 0 if i and j are adjacent, and Mij = 0 if i and j are nonadjacent, has the Strong Arnold Property: there is no nonzero real symmetric V ×V matrix X with MX = 0 and Xij = 0 whenever i and j are equal or adjacent. (A graph G is flat if it can be embedded injectively in 3-dimensional Euclidean space such that the image of any circuit is the boundary of some disk disjoint from the image of the remainder of the graph.) This applies to the Colin de Verdière graph parameter, and extends similar results for 2-connected outerplanar graphs and 3-connected planar graphs.