The Strong Arnold Property for 4connected flat graphs
Abstract
We show that if $G=(V,E)$ is a 4connected flat graph, then any real symmetric $V\times V$ matrix $M$ with exactly one negative eigenvalue and satisfying, for any two distinct vertices $i$ and $j$, $M_{ij}<0$ if $i$ and $j$ are adjacent, and $M_{ij}=0$ if $i$ and $j$ are nonadjacent, has the Strong Arnold Property: there is no nonzero real symmetric $V\times V$ matrix $X$ with $MX=0$ and $X_{ij}=0$ whenever $i$ and $j$ are equal or adjacent. (A graph $G$ is {\em 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 2connected outerplanar graphs and 3connected planar graphs.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.03200
 Bibcode:
 2015arXiv151203200S
 Keywords:

 Mathematics  Combinatorics;
 05C50;
 15A18;
 05C10