The Colin de Verdière number and graphs of polytopes
| dc.creator | Izmestiev, Ivan | |
| dc.date | 2007-04-03 | |
| dc.date | 2008-07-25 | |
| dc.date.accessioned | 2026-07-07T09:52:27Z | |
| dc.date.available | 2026-07-07T09:52:27Z | |
| dc.description | The Colin de Verdière number $μ(G)$ of a graph $G$ is the maximum corank of a Colin de Verdière matrix for $G$ (that is, of a Schrödinger operator on $G$ with a single negative eigenvalue). In 2001, Lovász gave a construction that associated to every convex 3-polytope a Colin de Verdière matrix of corank 3 for its 1-skeleton. We generalize the Lovász construction to higher dimensions by interpreting it as minus the Hessian matrix of the volume of the polar dual. As a corollary, $μ(G) \ge d$ if $G$ is the 1-skeleton of a convex $d$-polytope. Determination of the signature of the Hessian of the volume is based on the second Minkowski inequality for mixed volumes and on Bol's condition for equality. | |
| dc.description | 18 pages, 2 figures; reorganized; an estimate of the spectral gap added; the appendix on mixed volumes rewritten | |
| dc.identifier | https://arxiv.org/abs/0704.0349 | |
| dc.identifier | http://arxiv.org/abs/0704.0349 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/165601 | |
| dc.subject | Combinatorics | |
| dc.subject | Metric Geometry | |
| dc.subject | 05C50 (primary) 52A39, 52C25 (secondary) | |
| dc.title | The Colin de Verdière number and graphs of polytopes | |
| dc.type | text |