Magical Property of Fullerenes

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.