Magical Property of Fullerenes

TL;DR

This paper investigates a special numerical property of fullerenes, showing certain structures cannot have it, while others can, revealing new combinatorial characteristics of these carbon molecules.

Contribution

It introduces the concept of the magical property in fullerenes and characterizes which fullerene structures admit such arrangements, providing new insights into their combinatorial properties.

Findings

$C_{8n+4}$ fullerenes do not admit the magical property

Some fullerenes like $C_{24}$ and $C_{26}$ have many arrangements with the property

The paper establishes conditions for the existence of such arrangements

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.