The Colin de Verdière number and graphs of polytopes
Abstract
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.
18 pages, 2 figures; reorganized; an estimate of the spectral gap added; the appendix on mixed volumes rewritten
18 pages, 2 figures; reorganized; an estimate of the spectral gap added; the appendix on mixed volumes rewritten