The Generation of Fullerenes

We describe an efficient new algorithm for the generation of fullerenes. Our implementation of this algorithm is more than 3.5 times faster than the previously fastest generator for fullerenes -- fullgen -- and the first program since fullgen to be useful for more than 100 vertices. We also note a programming error in fullgen that caused problems for 136 or more vertices. We tabulate the numbers of fullerenes and IPR fullerenes up to 400 vertices. We also check up to 316 vertices a conjecture of Barnette that cubic planar graphs with maximum face size 6 are hamiltonian and verify that the smallest counterexample to the spiral conjecture has 380 vertices.Comment: 21 pages; added a not

The topology of fullerenes

Fullerenes are carbon molecules that form polyhedral cages. Their bond structures are exactly the planar cubic graphs that have only pentagon and hexagon faces. Strikingly, a number of chemical properties of a fullerene can be derived from its graph structure. A rich mathematics of cubic planar graphs and fullerene graphs has grown since they were studied by Goldberg, Coxeter, and others in the early 20th century, and many mathematical properties of fullerenes have found simple and beautiful solutions. Yet many interesting chemical and mathematical problems in the field remain open. In this paper, we present a general overview of recent topological and graph theoretical developments in fullerene research over the past two decades, describing both solved and open problems. WIREs Comput Mol Sci 2015, 5:96–145. doi: 10.1002/wcms.1207 Conflict of interest: The authors have declared no conflicts of interest for this article. For further resources related to this article, please visit the WIREs website

O kanonskom označavanju ugljikovih atoma u fullerenima: C60 buckminsterfulleren

Numbering of atoms in relatively large molecules, such as fullerenes appears for most part to be arbitrary or based on ad hoc schemes. We argue in favor of the use of a particular canonical labeling of atoms in molecules based on the smallest possible binary molecular code obtained from the adjacency matrix when its rows are read from left to right and from top to bottom. The approach has been illustrated with buckminsterfullerene. We have outlined advantages of the approach and have shown that finding canonical labels is practical even in the case of large regular graphs.Označavanje atoma u velikim i složenim molekulama vrlo je često proizvoljno ili je zasnovano na ad hoc shemama. Preporuča se novo i sustavno označavanje atoma u molekulama koje se temelji na najmanjem mogućem binarnom molekularnom kodu. Čitanje matrice susjedstva grafa molekule po retcima definira binarni molekularni kod: najmanji među njima se zove kanonski kod. Na primjeru molekule buckminsterfullerena pokazano je kako se može odrediti ovaj kod. Istaknute su prednosti novoga načina označavanja atoma u velikim molekulama a posebice za one prikazane regularnim grafovima

From C 58 to C 62 and back: Stability, structural similarity, and ring current

An increasing number of observations show that non-classical isomers may play an important role in the formation of fullerenes and their exo- and endo-derivatives. A quantum-mechanical study of all classical isomers of C58, C60, and C62, and all non-classical isomers with at most one square or heptagonal face, was carried out. Calculations at the B3LYP/6-31G* level show that the favored isomers of C58, C60, and C62 have closely related structures and suggest plausible inter-conversion and growth pathways among low-energy isomers. Similarity of the favored structures is reinforced by comparison of calculated ring currents induced on faces of these polyhedral cages by radial external magnetic fields, implying patterns of magnetic response similar to those of the stable, isolated-pentagon C60 molecule