The Colin de Verdi`ere number and graphs of polytopes

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.

💡 Research Summary

The paper investigates the relationship between the Colin‑de‑Verdière number μ(G) of a graph G and the combinatorial geometry of convex polytopes whose 1‑skeleton coincides with G. The Colin‑de‑Verdière number is defined as the maximum corank of a symmetric matrix M that satisfies three conditions: (i) M has exactly one negative eigenvalue, (ii) M is a Schrödinger‑type operator on the graph (i.e., off‑diagonal entries are non‑positive and vanish precisely when the corresponding vertices are non‑adjacent), and (iii) M satisfies a strong Arnold property that prevents non‑trivial kernel vectors from being supported on a proper subgraph. Historically, Lovász showed that for any convex 3‑polytope the matrix constructed from the polytope’s geometry yields μ(G)=3, establishing a direct link between three‑dimensional convexity and the graph invariant.

The authors extend Lovász’s construction to arbitrary dimension d≥3. Their key observation is that the Hessian of the volume of the polar dual polytope, when expressed as a function of the support parameters (the distances from the origin to the supporting hyperplanes), is, up to a sign, a Colin‑de‑Verdière matrix for the original polytope’s 1‑skeleton. Concretely, let P be a full‑dimensional convex polytope containing the origin in its interior, and let P* be its polar dual. If λ=(λ₁,…,λ_n) denotes the vector of support numbers associated with the facets of P (equivalently, the distances of the vertices of P* from the origin), the volume V(λ)=Vol(P*(λ)) is a smooth, homogeneous function of degree d. The matrix H with entries H_{ij}=∂²V/∂λ_i∂λ_j is symmetric, has zero row sums, and its off‑diagonal entries are non‑positive because increasing two distinct support parameters simultaneously reduces the volume more than increasing them separately. By taking M=−H, the authors obtain a matrix that satisfies the first two Colin‑de‑Verdière conditions automatically.

The delicate part of the argument is to prove that M has exactly one negative eigenvalue and that its kernel has dimension d. This is achieved by invoking two deep results from convex geometry. First, the second Minkowski inequality for mixed volumes provides a quadratic form inequality that guarantees the Hessian is positive semidefinite on the subspace orthogonal to the vector (1,…,1). Equality in this inequality occurs only under very special circumstances (when the involved convex bodies are homothetic). Second, Bol’s equality condition for the second Minkowski inequality is used to rule out the possibility of more than one negative direction: in a generic polytope the inequality is strict, ensuring that the Hessian has exactly one negative eigenvalue. Consequently, the corank of M equals d, the dimension of the polytope, and therefore μ(G)≥d for any graph G that is the 1‑skeleton of a convex d‑polytope. Moreover, when the polytope is in generic position, the inequality is strict and the authors obtain μ(G)=d.

The paper proceeds to discuss several corollaries and implications. The result provides a geometric proof of the known fact that planar graphs have μ≤3 (since a planar graph can be realized as the 1‑skeleton of a convex 3‑polytope) and extends it to higher dimensions, offering a new tool for bounding μ(G) from below using polyhedral realizations. The authors also remark that the Hessian‑based construction is more transparent than previous Laplacian‑based approaches because it directly reflects the curvature of the volume functional, and it behaves well under affine transformations of the polytope. Finally, the authors suggest future research directions, such as extending the method to non‑convex or non‑polytopal complexes, exploring connections with other spectral graph invariants (e.g., the algebraic connectivity), and investigating whether the Hessian of other geometric functionals (surface area, mean width) can yield Colin‑de‑Verdière matrices for broader classes of graphs.