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Some open mathematical problems on fullerenes

Artur Bille, Victor Buchstaber, Evgeny Spodarev

Abstract

Fullerenes are hollow carbon molecules where each atom is connected to exactly three other atoms, arranged in pentagonal and hexagonal rings. Mathematically, they can be combinatorially modeled as planar, 3-regular graphs with facets composed only of pentagons and hexagons. In this work, we outline a few of the many open questions about fullerenes, beginning with the problem of generating fullerenes randomly. We then introduce an infinite family of fullerenes on which the generalized Stone-Wales operation is inapplicable. Furthermore, we present numerical insights on a graph invariant, called \textit{character} of a fullerene, derived from its adjacency and degree matrices. This descriptor may lead to a new method for linear enumeration of all fullerenes.

Some open mathematical problems on fullerenes

Abstract

Fullerenes are hollow carbon molecules where each atom is connected to exactly three other atoms, arranged in pentagonal and hexagonal rings. Mathematically, they can be combinatorially modeled as planar, 3-regular graphs with facets composed only of pentagons and hexagons. In this work, we outline a few of the many open questions about fullerenes, beginning with the problem of generating fullerenes randomly. We then introduce an infinite family of fullerenes on which the generalized Stone-Wales operation is inapplicable. Furthermore, we present numerical insights on a graph invariant, called \textit{character} of a fullerene, derived from its adjacency and degree matrices. This descriptor may lead to a new method for linear enumeration of all fullerenes.

Paper Structure

This paper contains 13 sections, 1 theorem, 14 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Random fullerenes
  4. Naive approach
  5. Generating algorithms for fullerenes
  6. Face spiral acceptance-rejection algorithm
  7. Stones--Wales Operation
  8. Polyhedral Stone--Wales Operation
  9. Generalized Stone--Wales Operation
  10. Spectra of fullerenes
  11. Characters as unique fullerene descriptors
  12. Linear ordering of fullerenes and their limiting shape
  13. Summary

Key Result

Theorem 1

There exist infinitely many fullerenes on which a gSW operation cannot be applied.

Figures (10)

  • Figure 1: pSW operation on four vertices with arbitrary degrees.
  • Figure 2: A gSW operation on a gSW path with ten vertices ($w=5$) with thick edges representing the gSW path necessary for the gSW operation.
  • Figure 3: Left: General structure of a $t$-triangle. Right: A $(4,(1,0,0))$-triangle with deleted edges (dotted lines) and deleted vertices (labeled $X$). Thick lines represent open edges.
  • Figure 4: Cases of structures of $T_n^6$ with $2$-facet and $3$-facet vertices (labeled $2$ and $3$, respectively).
  • Figure 5: $T_{56,622}^6$ (upper left), $T_{80,31924}^6$ (upper right) and $T_{96,191839}^6$ (lower left) with $2$-facet vertices colored blue. The cut-partition of $T_{56,622}$ yields four $2$-triangles, and $T_{80,31924}$ yields 20 $1$-triangles. Applying the first part of Construction \ref{['con:cutedge-partition']} to $T_{96,191839}^6$ produces 12 $1$-triangles and two copies of the structure in the lower center. Second part of the construction decomposes each of these copies into a $3$-triangle and three $1$-triangles. Bold edges represent the path $\boldsymbol{p}$.
  • ...and 5 more figures

Theorems & Definitions (10)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Conjecture 1
  • Conjecture 2
  • Theorem 1
  • proof
  • Definition 5
  • Conjecture 3