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

Taizo Sadahiro

TL;DR

This work presents a non-abelian ${\mathfrak S}_3$-Galois cover of the complete graph $K_4$, realized as the star graph $X_3$, whose intermediate covers are the cube $Q$ and the truncated tetrahedron $T$. By exploiting the Ihara zeta-function framework and its $L$-function factorization for Galois covers, the author derives precise spectral relations among the covers, including ${\rm Spec}(Y) \cup {\rm Spec}(X) \cup {\rm Spec}(X) = {\rm Spec}(Q) \cup {\rm Spec}(T) \cup {\rm Spec}(T)$, and provides explicit eigenpolynomials for each graph. The geometry of these covers is tied to the honeycomb lattice, yielding a Fourier-analytic description of the spectra and revealing that $K_4$, $Q$, $T$, and $X_3$ arise as quotients of a common hexagonal lattice under specific crystallographic group actions. The results offer a cohesive Galois-theoretic interpretation of integral spectra for several small polyhedral graphs and motivate extensions to larger symmetry groups and their intermediate covers.

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.

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

TL;DR

This work presents a non-abelian -Galois cover of the complete graph , realized as the star graph , whose intermediate covers are the cube and the truncated tetrahedron . By exploiting the Ihara zeta-function framework and its -function factorization for Galois covers, the author derives precise spectral relations among the covers, including , and provides explicit eigenpolynomials for each graph. The geometry of these covers is tied to the honeycomb lattice, yielding a Fourier-analytic description of the spectra and revealing that , , , and arise as quotients of a common hexagonal lattice under specific crystallographic group actions. The results offer a cohesive Galois-theoretic interpretation of integral spectra for several small polyhedral graphs and motivate extensions to larger symmetry groups and their intermediate covers.

Abstract

We show that the star graph defined as the Cayley graph of generated by the star transpositions is an -cover of the complete graph , which is known to have fine spectral properties. In the case , 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.

Paper Structure

This paper contains 5 sections, 12 theorems, 54 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Construction of ${\mathfrak S}_n$-cover of $K_{n+1}$
  3. Spectral properties
  4. Geometric properties and Fourier analysis
  5. Concluding comments

Key Result

Theorem 1

Let $\tau_{i}=(i, n+1)\in {\mathfrak S}_{n+1}$ be the transposition for $i=1,2,\ldots,n$, and let $X_n={\rm Cay}({\mathfrak S}_{n+1}, \{\tau_1, \tau_2, \ldots, \tau_n\})$. Then, $X_n$ is an ${\mathfrak S}_n$-cover of the complete graph $K_{n+1}$.

Figures (6)

  • Figure 1: A periodic boundary condition with hexagonal fundamental domain which make the adjacency operator on the honeycomb lattice to have only integral eigenvalues.
  • Figure 2: Left: $X_3={\rm Cay}({\mathfrak S}_4,\{(1,4),(2,4),(3,4)\})$. Right: The orbits of the right action by the subgroup $\langle(1,2)\rangle\in G_3\cong {\mathfrak S}_3$ makes a quotient graph of $X_3$ which is isomorphic to the truncated tetrahedron graph.
  • Figure 3: The orbits of the right action by the subgroup $\langle(1,2,3)\rangle$ make a quotient graph of $X_3$ which is isomorphic to the cube graph. The vertex marked $i$ represents the orbit containing the even permutation $\xi\in {\mathfrak S}_{4}$ with $\xi(4)=i$, and that marked $i$ corresponds to the odd permutations.
  • Figure 4: $\Lambda_{X_3}$ and $\Lambda_Q$
  • Figure 5: Fundamental domain for the action of $G_{K_4}$ (left) and $\Lambda_{Q}$ (right)
  • ...and 1 more figures

Theorems & Definitions (25)

  • Definition 1: Graph Cover
  • Definition 2: Galois Cover
  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2: Krakovski and Mohar krakovski2012spectrum
  • Theorem 3: Chapuy and Féray ChapuyFeray
  • Definition 3: Intermediate cover
  • Theorem 4: Galois correspondence for graph covers terras2010zeta
  • Theorem 5
  • ...and 15 more