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

TL;DR

This paper explores the spectral and geometric properties of star graphs as covers of complete graphs, revealing new insights into their embeddings, spectra, and connections to polyhedral graphs.

Contribution

It demonstrates that star graphs are covers of complete graphs with notable spectral and geometric properties, including embeddings into lattices and explicit spectrum calculations.

Findings

Star graphs are ${ m S}_n$-covers of complete graphs with fine spectral properties.

For n=3, the star graph embeds into the honeycomb lattice and has a computable spectrum.

Intermediate covers relate to cube and truncated tetrahedron, explaining their integral spectra.

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.