The Colin de Verdière number and graphs of polytopes

Details The Colin de Verdière number and graphs of polytopes The Colin de Verdière number and graphs of polytopes

2007-04-03

arxiv.org/abs/0704.0349v3

Mathematics

Combinatorics

18 pages, 2 figures; reorganized; an estimate of the spectral gap added; the appendix on mixed volumes rewritten

Scientific paper

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.

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