Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Friday, May 20, 2016, 12:15 pm

Duration: 30 minutes

Location: CAB G11

Speaker: Carsten Lange (TU München)

Cayley gaphs of finite Coxeter groups, Colin de Verdiere matrices and convex polytopes

In 2001, Lovasz explicitly described how Colin de Verdiere matrices of a 3-connected planar graph G correspond to representations of G as 1-skeleton of a convex 3-polytope. The proof extends an earlier result of Lovasz and Schrijver and exploits spectral properties of these matrices associated to G. We now consider the following open problem for d>3. Suppose we are given a Colin de Verdiere matrix M for a graph G that is known to be the 1-skeleton of some convex d-polytope although the polytope is not known. Is it then possible to recover from M a convex d-polytope with 1-skeleton G? In my talk, I wil explain the notion of a Colin de Verdiere matrix, outline the approach of Lovasz in dimension 3 and indicate a positive answer of the problem for the Cayley graph of any finite Coxeter group.

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