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arXiv · 2508.18593

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

Taizo Sadahiro
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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.

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Taizo Sadahiro. 2026-02-05. An ${\mathfrak S}_3$-cover of $K_4$ and integral polyhedral graphs. https://arxiv.org/abs/2508.18593

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