Journal of Mathematical Chemistry

, Volume 55, Issue 8, pp 1669–1682 | Cite as

Sizes of pentagonal clusters in fullerenes

  • Nino Bašić
  • Gunnar Brinkmann
  • Patrick W. Fowler
  • Tomaž Pisanski
  • Nico Van Cleemput
Original Paper


Stability and chemistry, both exohedral and endohedral, of fullerenes are critically dependent on the distribution of their obligatory 12 pentagonal faces. It is well known that there are infinitely many IPR-fullerenes and that the pentagons in these fullerenes can be at an arbitrarily large distance from each other. IPR-fullerenes can be described as fullerenes in which each connected cluster of pentagons has size 1. In this paper we study the combinations of cluster sizes that can occur in fullerenes and whether the clusters can be at an arbitrarily large distance from each other. For each possible partition of the number 12, we are able to decide whether the partition describes the sizes of pentagon clusters in a possible fullerene, and state whether the different clusters can be at an arbitrarily large distance from each other. We will prove that all partitions with largest cluster of size 5 or less can occur in an infinite number of fullerenes with the clusters at an arbitrarily large distance of each other, that 9 partitions occur in only a finite number of fullerene isomers and that 15 partitions do not occur at all in fullerenes.


Fullerene Patch Distance Pentagonal incidence partition 

Mathematics Subject Classification

05C10 52B10 92E10 



The work of T.P. and N.B has been supported in part by ARRS Slovenia, Projects: P1-0294, N1-0032. In addition the work of T.P. was supported by Projects J1-7051 and J1-6720.


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Copyright information

© Springer International Publishing Switzerland 2017

Authors and Affiliations

  1. 1.University of PrimorskaKoperSlovenia
  2. 2.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia
  3. 3.Applied Mathematics, Computer Science and StatisticsGhent UniversityGentBelgium
  4. 4.Department of ChemistryUniversity of SheffieldSheffieldUK

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