A spectral graph realization is an embedding of a finite simple graph into Euclidean space that is constructed from the eigenvalues and eigenvectors of the graph's adjacency matrix. It has previously been observed that some polytopes can be reconstructed from their edge-graphs by taking the convex hull of a spectral realization of this edge-graph. These polytopes, which we shall call spectral polytopes, have remarkable rigidity and symmetry properties and are a source for many open questions.
In this thesis we aim to further the understanding of this phenomenon by exploring the geometric and combinatorial properties of spectral polytopes on several levels.
One of our central questions is whether already weak forms of symmetry can be a sufficient reason for a polytope to be spectral. To answer this, we derive a geometric criterion for the identification of spectral polytopes and apply it to prove that indeed all polytopes of combined vertex- and edge-transitivity are spectral, admit a unique reconstruction from the edge-graph and realize all the symmetries of this edge-graph. We explore the same questions for graph realizations and find that realizations of combined vertex- and edge-transitivity are not necessarily spectral.
Instead we show that we require a stronger form of symmetry, called distance-transitivity.
Motivated by these findings we take a closer look at the class of edge-transitive polytopes, for which no classification is known. We state a conjecture for a potential classification and provide complete classifications for several sub-classes, such as distance-transitive polytopes and edge-transitive polytopes that are not vertex-transitive. In particular, we show that the latter class contains only polytopes of dimension d <= 3.
As a side result we obtain the complete classification of the vertex-transitive zonotopes and a new characterization for root systems.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa:de:qucosa:75215 |
| Date | 29 June 2021 |
| Creators | Winter, Martin |
| Contributors | Helmberg, Christoph, Joswig, Michael, Schulte, Egon, Technische Universität Chemnitz |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | info:eu-repo/semantics/publishedVersion, doc-type:doctoralThesis, info:eu-repo/semantics/doctoralThesis, doc-type:Text |
| Rights | info:eu-repo/semantics/openAccess |
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