An ${\mathfrak{S}}_3$-cover of $K_4$ and integral polyhedral graphs

Abstract

We show that the star graph defined as the Cayley graph of ${\mathfrak{S}}_{n+1}$ generated by the star transpositions is an ${\mathfrak{S}}_n$-cover of the complete graph $K_{n+1}$, which is known to have fine spectral properties. In the case $n=3$, the star graph also has fine geometric properties: it embeds into the honeycomb lattice and has a spectrum computable via both representation theory and an explicit Fourier formula. Intermediate covers correspond to the cube and truncated tetrahedron, offering a new interpretation of their integral spectra.