Mark Ramras, Brigitte Servatius, Andrei V. Zelevinsky
Date of Award
Doctor of Philosophy
Department or Academic Unit
College of Arts and Sciences. Department of Mathematics.
This dissertation deals with highly symmetric abstract polytopes. Abstract polytopes are combinatorial structures which generalize the classical notion of convex polytopes. There are many different natural graphs associated with an abstract polytope. Particular attention is given to two of these graphs, namely the comparability graph of a polytope and the Hasse diagram of a polytope. We study various types of transitivity in these graphs.
Both the comparability graph and the Hasse diagram for a polytope inherit nicely the rank function associated with a polytope. Using this rank function, we can study specific subgraphs of each graph, by restricting to vertices in a certain range of ranks. If a polytope is of even rank, there is a natural notion of a \medial layer graph," which is the restriction of either the comparability graph, or the Hasse diagram to the middle two ranks. However, if a polytope is of odd rank, and we restrict to the middle three ranks, then we obtain two different graphs, one from the comparability graph, and one from the Hasse diagram.
We study various transitivity properties of these classes of graphs associated with polytopes. We are able to establish polytope properties that are necessary for a graph to have certain transitivities. And conversely, if we require a medial layer graph to have some transitivity, we can understand combinatorial properties of the associated polytope.
In particular, Monson andWeiss proved a theorem in 2005 about arc transitivity of medial layer graphs of 4-polytopes; we prove an analog of that theorem for polytopes of arbitrary even rank. In odd rank, transitivity of the two possible medial layer graphs had not been studied extensively in the past. We provide some results for polytopes of arbitrary rank, as well as a classiffication for polytopes of rank 3.
Mixer, Mark, "Transitivity of graphs associated with highly symmetric polytopes" (2010). Mathematics Dissertations. Paper 11. http://hdl.handle.net/2047/d20000244
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