Magic property of fullerenes

Abstract

Fullerenes are an allotrope of carbon having a hollow, cage-like structure. The atoms in the molecule are arranged in pentagonal and hexag- onal 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 a magic property if the numbers 1, 2, . . . , n may be assigned to its vertices so that the sums of the numbers on each pentagonal face are equal and the sums of the numbers in each hexagonal face are equal. We show that C8n+4 does not admit such an arrangement for all n, while there are fullerenes, like C24 and C26 that have many nonisomorphic such arrangements.