The Colin de Verdi\`ere number and graphs of polytopes

TL;DR

This paper extends Lovász's construction linking the Colin de Verdière number to convex polytopes, showing that the number is at least the dimension for graphs of convex polytopes, using volume Hessians and Minkowski inequalities.

Contribution

It generalizes Lovász's 3D polytope construction to higher dimensions by relating the Colin de Verdière number to the volume Hessian of polar duals.

Findings

The Colin de Verdière number is at least the polytope dimension.

The construction uses the Hessian of the volume of the polar dual.

The signature of the Hessian is determined by Minkowski inequalities.

Abstract

The Colin de Verdi\`ere number $\mu(G)$ of a graph $G$ is the maximum corank of a Colin de Verdi\`ere matrix for $G$ (that is, of a Schr\"odinger operator on $G$ with a single negative eigenvalue). In 2001, Lov\'asz gave a construction that associated to every convex 3-polytope a Colin de Verdi\`ere matrix of corank 3 for its 1-skeleton. We generalize the Lov\'asz construction to higher dimensions by interpreting it as minus the Hessian matrix of the volume of the polar dual. As a corollary, $\mu(G) \ge d$ 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.