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Home > Seminars > 2014

The topology of fullerenes

Peter Schwerdtfeger, Massey University (Auckland, Nouvelle-Zélande)

Séminaire LCPQ

Salle de séminaire IRSAMC, 14h – 15h

Fullerenes are 3-connected cubic planar graphs consisting of pentagons and hexagons only. There has been great progress over the last two decades describing the topological and graph theoretical properties of fullerenes, but leaving still many unsolved and interesting mathematical (and chemical) problems open in this field. A few example are, i) how to generate all possible non-isomorphic graphs for a fixed vertex count, ii) are fullerenes Hamiltonian (Barnette’s conjecture) and what is the number of distinct Hamiltonian cycles, iii) the Pauling bond order and the number of perfect matchings, iv) the search for suitable topological indices to find the most stable fullerene structure out of the many (N9) possibilities, or how to pack fullerene cages in 3D space (Hilbert problem)? Our research group in Albany is developing a general-purpose program (Program Fullerene) [1,2] that creates 2D graphs and accurate 3D structures for any fullerene isomer through various different graph-theoretical methods and algorithms, and subsequently performs a topological analysis. A general overview on topological and graph theoretical aspects of fullerenes is presented, and illustrated for many different fullerenes ranging from N = 20 to 20,000 vertices, and some new conjectures and theorems are presented.


[1] P. Schwerdtfeger, L. Wirz, J. Avery, J. Comput. Chem. 34, 1508-1526 (2013).

[2] O. Ori, M. V. Putz, I. Gutman, P. Schwerdtfeger, “Generalized Stone-Wales Transformations for Fullerene Graphs Derived from Berge’s Switching Theorem”, submitted.

[3] L. Wirz, J. Avery, P. Schwerdtfeger, “Structure and Properties of the Non-Spiral Fullerenes T-C380, D3-C384, D3-C440 and D3-C672 and their Halma and Leapfrog Transforms”, submitted.