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

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.