I consider *convex polytopes* $P\subseteq\Bbb R^d$ (convex hull of finitely many points) which are arc-transitive, i.e. where the automorphism group acts transitively on the 1-flags (incident vertex-edge pairs). Especially, this includes that the polytope is vertex- and edge-transitive. The *graph* of a polytope is the graph isomorphic to its 1-skeleton.



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> **Question**: Are there arc-transitive polytopes, where the graph has more symmetries than the polytope?
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When weakening the requirement of arc-transitivity, there are examples:


* A rhombus is edge- but not vertex-transitive. However, its graph is vertex-transitive. (Thanks to Henrik for the comment)
* There are vertex-transitive neighborly polytopes other than a simplex, but none of these can be edge-transitive. Their graphs are complete and are therefore edge-transitive.



Can the graph of a symmetric polytope have more symmetries than the polytope itself?
The answer is *No*, the graph of an arc-transitive polytope cannot have more symmetries than the polytope.
The polytope and its graph have the exact same symmetries!




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In the article [*"Capturing Polytopal Symmetries by Coloring the Edge-Graph"*](https://arxiv.org/abs/2108.13483) I prove the following:



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> (Corollary 5.3.) Let $P\subset \Bbb R^d$ be a polytope with edge-graph $G$. Consider a coloring $G^c$ of the edge-graph where two vertices/edges have the same color if and only if they are in the same orbit w.r.t. the symmetry group $\mathrm{Aut}(P)\subseteq\mathrm O(\Bbb R^d)$. Then $\mathrm{Aut}(G^c)\simeq\mathrm{Aut}(P)$.
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In particular, if $P$ is vertex- and edge-transitive, then each vertex/edge of $G^c$ has the same color, that is, the graph is uncolored. Thus $\mathrm{Aut}(G)\simeq\mathrm{Aut}(P)$.


At the core of the proof is a result by Ivan Izmestiev from



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> [I. Izmestiev (2010), *"The Colin de Verdière Number and Graphs of Polytopes"*](https://link.springer.com/article/10.1007/s11856-010-0070-5)
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(summarized as Theorem 4.1 in my article), which roughly states that polytope skeleta are *spectral embeddings* of the edge-graph, when assigning certain vertex/edge weights.
These weights can be used as colors, and it then follows from basic spectral graph theory that every combinatorial symmetry of the colored graph extends to a geometric symmetry of the polytope.


