Fullerene Notation and Isomerization Operations

All Pairs of Pentagons in Leapfrog Fullerenes Are Nice

A subgraph H of a graph G with perfect matching is nice if G−V(H) has perfect matching. It is well-known that all fullerene graphs have perfect matchings and that all fullerene graphs contain some small connected graphs as nice subgraphs. In this contribution, we consider fullerene graphs arising from smaller fullerenes via the leapfrog transformation, and show that in such graphs, each pair of (necessarily disjoint) pentagons is nice. That answers in affirmative a question posed in a recent paper on nice pairs of odd cycles in fullerene graphs.

Scratching the surface of buckminsterfullerene: the barriers for Stone-Wales transformation through symmetric and asymmetric transition states

General-gradient approximation (PBE) and hybrid Hartree-Fock density functional theories (B3LYP) in conjunction with basis sets of up to polarized triple-zeta quality have been applied to study the Stone-Wales transformation of buckminsterfullerene (BF) to yield a C(60) isomer of C(2)(v) symmetry with two adjacent pentagons (#1809). In agreement with earlier investigations, two different transition states and reaction pathways could be identified for the rearrangement from BF to C(60)-C(2)(v) on the C(60) potential energy surface (PES). One has C(2) molecular point group symmetry with the two migrating carbon atoms remaining close to the fullerene surface. The other one has a high-energy carbene-like (sp(3)) structure where a single carbon atom is significantly moved away from the C(60) surface. The carbene intermediate and the second transition state along the stepwise reaction path characterized previously at lower levels of theory do not exist as stationary points with the density functionals utilized here. The classical barriers of both mechanisms are essentially identical, 6.9 eV using PBE and 7.3 eV with B3LYP.

Planar rearrangements of fullerenes

A notion of planar rearrangement of fullerenes is proposed as a general framework for all conceivable rearrangements of fullerenes satisfying a minimum physicochemical condition. The planar rearrangement is defined as an edge relocation such that the planarity of a given Schlegel diagram remains undisturbed. Graph-theoretical properties of planar rearrangements are discussed and characteristics for establishing a hierarchy between them are pointed out. A number of graph-theoretical concepts have been introduced to provide means for classification and systematic generation of fullerene rearrangements. The simpliest nontrivial rearrangements is shown to be the Stone-Wales rearrangement.