Magical Property of Fullerenes
Djordje Baralic, Adam Farhat
TL;DR
The paper investigates when a fullerene graph $C_n$ admits a magic configuration, i.e., a bijection $f:V\to\{1,\dots,n\}$ that makes all pentagonal face sums equal to $S_p$ and all hexagonal face sums equal to $S_h$, constrained by $24 S_p+(n-20) S_h=3n(n+1)$. It proves a modular obstruction ($n\equiv4\pmod{8}$ implies no magic configuration) and then performs exhaustive CSP-CP-SAT enumerations for the small fullerenes $C_{24}$ and $C_{26}$, identifying all valid labelings for the admissible $(S_p,S_h)$ pairs. The study shows the configuration space carries a free action by the fullerene automorphism group and a $\mathbf{Z}_2$ symmetry, enabling a symmetry-aware counting of nonisomorphic configurations, with detailed results for $C_{24}$ and $C_{26}$. Finally, it uses Principal Component Analysis to visualize the high-dimensional solution space, revealing a structured, symmetry-driven, low-dimensional organization of magic configurations and confirming invariances of the covariance under certain constant-transformations.
Abstract
Fullerenes are an allotrope of carbon having hollow, cage-like structure. Atoms in the molecule are arranged in pentagonal and hexagonal rings, such that each atom is connected to three other atoms. Simple polyhedra having only pentagonal and hexagonal faces are a mathematical model for fullerenes. We say that a fullerene with $n$ vertices has magical property if the numbers $1, 2, \dots, n$ may be assigned to its vertices so that the sums of the numbers in each pentagonal faces are equal and the sums of the numbers in each hexagonal faces are equal. We show that $C_{8n+4}$ does not admit such an arrangement for all $n$, while there are fullerenes, like $C_{24}$ and $C_{26}$ that have many nonisomorphic such arrangements.