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 Sn+1 generated by the star transpositions is an Sn -cover of the complete graph Kn+1 , which is known to have fine spectral properties. In the case n = 3, the star graph also has fine geometric properties: it em- beds into the honeycomb lattice and has a spectrum computable via both representation theory and an explicit Fourier formula. Inter- mediate covers correspond to the cube and truncated tetrahedron, offering a new interpretation of their integral spectra