Why are the 1-skeleton graphs of the Regular Polytopes distance transitive?

Question

A graph $G$ is distance transitive if for all vertices $u,v,w,x$ of $G$ such that $D_G(u,v) = D_G(w,x) $ implies that there exists a graph automorphism of $G$, $\psi \in \Gamma(G)$ such that $\psi(u) = w$, $\, \psi(v) = x$.

If we have a regular polytope, $\mathcal{P}$, (spherical, euclidean or hyperbolic) then if we take the graph whose vertices are the vertices of $\mathcal{P}$ and whose vertices are adjacent if and only if they share an edge in $\mathcal{P}$.

I should be clear that I do NOT know that this s true for all regular polytopes. But checking the few I can they seem to be so.

Answer 1

As stated by Chris Godsil in the comments, not all Platonic solids are distance-transitive: the 24-cell, 120-cell ans 600-cell are not distance-transitive.

Here is a complete list of polytopes that are distance-transitive (i.e. not only their edge-graph is distance-transitive, but also their Euclidean symmetry group acts distance-transitively on the skeleton):

• regular polygons,
• the regular dodecahedron and regular icosahedron,
• the $d$-dimensional crosspolytope,
• cartesian powers of regular simplices (this includes, of course, the regular simplices, but also the regular $d$-dimensional cube as the power $[0,1]\times\cdots\times[0,1]$),
• hyper-simplices,
• demi-cubes,
• the $2_{21}$-polytope,
• the $3_{21}$-polytope.

See also Theorem 5.10 in my recent preprint:

• M. Winter, "Eigenpolytopes, Spectral Polytopes and Edge-Transitivity." arXiv preprint arXiv:2009.02179 (2020).

However, the true reason for why exactly those polytopes are distance-transitive is more complicated. Chris Godsil should indeed be the perfect person to explain this, as it was essentially proven in his paper

• C.D. Godsil, "Eigenpolytopes of distance regular graphs." Canadian Journal of Mathematics 50.4 (1998): 739-755.