The Colin de Verdière number and graphs of polytopes
The Colin de Verdière number and graphs of polytopes
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Colin de Verdiere 数和多胞体图
DOI：
10.1007/s11856-010-0070-5
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发表时间：
2007
影响因子：
1
通讯作者：
Ivan Izmestiev
中科院分区：
文献类型：
--
作者：
Ivan Izmestiev
图G的Colin de Verdière数μ（G）是G的Colin de Verdière矩阵的最大余秩（即G上具有单个负特征值的薛定谔算子的最大余秩）。2001年，Lovász给出了一个构造，它将每个凸3-多面体的1-骨架对应于一个corank为3的Colin de Verdière矩阵，我们将Lovász构造推广到高维，将其解释为极对偶的体积的Hessian矩阵的负。作为推论，μ（G）≥ d，如果G是凸d-多面体的1-骨架.确定体积的Hessian特征是基于混合体积的第二Minkowski不等式和Bol的相等条件.
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) ≥ 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.