On decompositions of leapfrog fullerenes

It is shown that given a fullerene F with the number of vertices n divisible by 4, and such that no two pentagons in F share an edge, the corresponding leapfrog fullerene Le(F) contains a long cycle of length 3n − 6 missing out only one hexagon.     

On lower bounds of number of perfect matchings in fullerene graphs

Couting perfect matchings in graphs is a very difficult problem. Some recently developed decomposition techniques allowed us to estimate the lower bound of the number of perfect matchings in certain classes of graphs. By applying these techniques, it will be shown that every fullerene graph with p vertices contains at least p/2+1 perfect matchings. It is a significant improvement over a previously published estimate, which claimed at least three perfect matchings in every fullerene graph. As an interesting chemical consequence, it is noted that every bisubstituted derivative of a fullerene still permits a Kekulé structure.     

Long cycles in fullerene graphs

It is conjectured that every fullerene graph is hamiltonian. Jendrol’ and Owens proved [J. Math. Chem. 18 (1995), pp. 83–90] that every fullerene graph on n vertices has a cycle of length at least 4n/5. In this paper we, improve this bound to 5n/6 − 2/3.     

Longest cycles in generalized Buckminsterfullerene graphs

It is shown that every regular 3-valent polyhedral graph whose faces are all 5-gons and 6-gons contains a cycle through at least 4/5 of its vertices.     

Nonbonding orbitals in fullerenes: nuts and cores in singular polyhedral graphs

A zero eigenvalue in the spectrum of the adjacency matrix of the graph representing an unsaturated carbon framework indicates the presence of a nonbonding pi orbital (NBO). A graph with at least one zero in the spectrum is singular; nonzero entries in the corresponding zero-eigenvalue eigenvector(s) (kernel eigenvectors) identify the core vertices. A nut graph has a single zero in its adjacency spectrum with a corresponding eigenvector for which all vertices lie in the core. Balanced and uniform trivalent (cubic) nut graphs are defined in terms of (-2, +1, +1) patterns of eigenvector entries around all vertices. In balanced nut graphs all vertices have such a pattern up to a scale factor; uniform nut graphs are balanced with scale factor one for every vertex. Nut graphs are rare among small fullerenes (41 of the 10 190 782 fullerene isomers on up to 120 vertices) but common among the small trivalent polyhedra (62 043 of the 398 383 nonbipartite polyhedra on up to 24 vertices). Two constructions are described, one that is conjectured to yield an infinite series of uniform nut fullerenes, and another that is conjectured to yield an infinite series of cubic polyhedral nut graphs. All hypothetical nut fullerenes found so far have some pentagon adjacencies: it is proved that all uniform nut fullerenes must have such adjacencies and that the NBO is totally symmetric in all balanced nut fullerenes. A single electron placed in the NBO of a uniform nut fullerene gives a spin density distribution with the smallest possible (4:1) ratio between most and least populated sites for an NBO. It is observed that, in all nut-fullerene graphs found so far, occupation of the NBO would require the fullerene to carry at least 3 negative charges, whereas in most carbon cages based on small nut cubic polyhedra, the NBO would be the highest occupied molecular orbital (HOMO) for the uncharged system.     

New Lower Bound on the Number of Perfect Matchings in Fullerene Graphs

A fullerene graph is a cubic and 3-connected plane graph (or spherical map) that has exactly 12 faces of size 5 and other faces of size 6, which can be regarded as the molecular graph of a fullerene. T. Doli [3] obtained that a fullerene graph with p vertices has at least (p+2)/2 perfect matchings by applying the recently developed decomposition techniques in matching theory of graphs. This note gets a better lower bound 3(p+2)/4 of the number of perfect matchings of a fullerene graph by finding its 2-extendability. This property further implies a chemical consequence that every derivative of a fullerene by substituting any two pairs of adjacent carbon atoms permits a Kekulé structure.     

Synthesis of a uranyl persulfide complex and quantum chemical studies of formation and topologies of hypothetical uranyl persulfide cage clusters

The compound Na(4)[(UO(2))(S(2))(3)](CH(3)OH)(8) was synthesized at room temperature in an oxygen-free environment. It contains a rare example of the [(UO(2))(S(2))(3)](4-) complex in which a uranyl ion is coordinated by three bidentate persulfide groups. We examined the possible linkage of these units to form nanoscale cage clusters analogous to those formed from uranyl peroxide polyhedra. Quantum chemical calculations at the density functional and multiconfigurational wave function levels show that the uranyl-persulfide-uranyl, U-(S(2))-U, dihedral angles of model clusters are bent due to partial covalent interactions. We propose that this bent interaction will favor assembly of uranyl ions through persulfide bridges into curved structures, potentially similar to the family of nanoscale cage clusters built from uranyl peroxide polyhedra. However, the U-(S(2))-U dihedral angles predicted for several model structures may be too tight for them to self-assemble into cage clusters with fullerene topologies in the absence of other uranyl-ion bridges that adopt a flatter configuration. Assembly of species such as [(UO(2))(S(2))(SH)(4)](4-) or [(UO(2))(S(2))(C(2)O(4))(4)](4-) into fullerene topologies with ~60 vertices may be favored by use of large counterions.     

Fullerene Graphs with Exponentially Many Perfect Matchings

We show that for all sufficiently large even p there is a fullerene graph on p vertices that has exponentially many perfect matchings in terms of the number of vertices. Further, we show that all fullerenes with full icosahedral symmetry group have exponentially many perfect matchings and indicate how such results could be extended to the fullerenes with lower symmetry.     

Cyclical Edge-Connectivity of Fullerene Graphs and (k, 6)-Cages

It is shown that every fullerene graph G is cyclically 5-edge-connected, i.e., that G cannot be separated into two components, each containing a cycle, by deletion of fewer than five edges. The result is then generalized to the case of (k,6)-cages, i.e., polyhedral cubic graphs whose faces are only k-gons and hexagons. Certain linear and exponential lower bounds on the number of perfect matchings in such graphs are also established.     

Simple Synthesis of a Chlorin - Fullerene Dyad with a Novel Ring-Closure Reaction

To model the photosynthetic reaction center, the chlorin - fullerene dyad 1 was synthesized in a one-pot reaction from a linear tetrapyrrole and C(60). It is the first dyad of this type which contains a fullerene and a chlorin unit, the chromophore of naturally occurring photosynthetic systems.     

A note on the cyclical edge-connectivity of fullerene graphs

A fullerene graph is a 3-regular (cubic) and 3-connected spherical graph that has exactly 12 pentagonal faces and other hexagonal faces. The cyclical edge-connectivity of a graph G is the maximum integer k such that G cannot be separated into two components, each containing a cycle, by deletion of fewer than k edges. Došlić proved that the cyclical edge-connectivity of every fullerene graph is equal to 5. By using Euler’s formula, we give a simplified proof, mending a small oversight in Došlić’s proof. Further, it is proved that the cyclical connectivity of every fullerene graph is also equal to 5.     

Competitive conversions of carbonyl-containing methanofullerenes induced by electron transfer

The electrochemical reduction of carbonyl-containing methanofullerenes is studied by methods of cyclic voltammetry and ESR with in situ electrolysis in the environment of 0.1 M Bu₄NBF₄ in o-dichlorobenzene-dimethylformamide (3:1 by volume). It is shown that the one-electron transfer onto the fullerene sphere of these compounds induces opening of the three-membered cycle, which leads to processes of retrocyclopropanation, transfer of the adduct onto the fullerene sphere of another molecule with the formation of bismethanofullerenes, and transformation into fullerodihydrofuran. The totality of competitive processes and the balance between these is defined by the nature of substituents at the exocarbon atom and the experimental conditions. The opening process is interpreted as the intramolecular electron transfer from the fullerene sphere onto the three-membered cycle.     

Interconnection networks for parallel molecular dynamics simulation based on hamiltonian cubic symmetric topology

A class of interconnection networks for efficient parallel MD simulations based on hamiltonian cubic symmetric graphs is presented. The cubic symmetric graphs have many desirable properties as interconnection networks since they have a low degree and are vertex- and edge-transitive. We present a method for scheduling collective communication routines that are used in parallel MD and are based on the property that the graphs in question have a Hamilton cycle, that is, a cycle going through all vertices of the graph. Analyzing these communication routines shows that hamiltonian cubic symmetric graphs of small diameter are good candidates for a topology that gives rise to an interconnection network with excellent properties, allowing faster communication and thus speeding up parallel MD simulation.     

An Upper Bound for the Clar Number of Fullerene Graphs

A fullerene graph is a three-regular and three-connected plane graph exactly 12 faces of which are pentagons and the remaining faces are hexagons. Let F _n be a fullerene graph with n vertices. The Clar number c(F _n ) of F _n is the maximum size of sextet patterns, the sets of disjoint hexagons which are all M-alternating for a perfect matching (or Kekulé structure) M of F _n . A sharp upper bound of Clar number for any fullerene graphs is obtained in this article: . Two famous members of fullerenes C₆₀ (Buckministerfullerene) and C₇₀ achieve this upper bound. There exist infinitely many fullerene graphs achieving this upper bound among zigzag and armchair carbon nanotubes.     

Thermally stable perfluoroalkylfullerenes with the skew-pentagonal-pyramid pattern: C60(C2F5)4O, C60(CF3)4O, and C60(CF3)6

Reaction of C(60) with CF(3)I at 550 degrees C, which is known to produce a single isomer of C(60)(CF(3))(2,4,6) and multiple isomers of C(60)(CF(3))(8,10), has now been found to produce an isomer of C(60)(CF(3))(6) with the C(s)-C(60)X(6) skew-pentagonal-pyramid (SPP) addition pattern and an epoxide with the C(s)-C(60)X(4)O variation of the SPP addition pattern, C(s)-C(60)(CF(3))(4)O. The structurally similar epoxide C(s)-C(60)(C(2)F(5))(4)O is one of the products of the reaction of C(60) with C(2)F(5)I at 430 degrees C. The three compounds have been characterized by mass spectrometry, DFT quantum chemical calculations, Raman, visible, and (19)F NMR spectroscopy, and, in the case of the two epoxides, single-crystal X-ray diffraction. The compound C(s)-C(60)(CF(3))(6) is the first [60]fullerene derivative with adjacent R(f) groups that are sufficiently sterically hindered to cause the (DFT-predicted) lengthening of the cage (CF(3))C-C(CF(3)) bond to 1.60 A as well as to give rise to a rare, non-fast-exchange-limit (19)F NMR spectrum at 20 degrees C. The compounds C(s)-C(60)(CF(3))(4)O and C(s)-C(60)(C(2)F(5))(4)O are the first poly(perfluoroalkyl)fullerene derivatives with a non-fluorine-containing exohedral substituent and the first fullerene epoxides known to be stable at elevated temperatures. All three compounds demonstrate that the SPP addition pattern is at least kinetically stable, if not thermodynamically stable, at temperatures exceeding 400 degrees C. The high-temperature synthesis of the two epoxides also indicates that perfluoroalkyl substituents can enhance the thermal stability of fullerene derivatives with other substituents.     

On bicriticality of (3,6)-fullerene graphs

A (3,6)-fullerene is a plane cubic graph whose faces are only triangles and hexagons. A connected graph G with at least \(2n+2\) vertices is said to be n-extendable if it has n independent edges and every set of n independent edges extends to a perfect matching of G. A graph G is said to be bicritical if for every pair of distinct vertices u and v, \(G-u-v\) has a perfect matching. It is known that every (3,6)-fullerene is 1-extendable, but not 2-extendable. In this short paper, we show that a (3,6)-fullerene G is bicritical if and only if G has the connectivity 3 and is not isomorphic to one graph (2,4,2). As a surprising consequence we have that a (3,6)-fullerene is bicritical if and only if each hexagonal face is resonant.     

Studies on the encapsulation of various anions in different fullerenes using density functional theory calculations and Born-Oppenheimer molecular dynamics simulation

The density functional theory (DFT)-based Becke's three parameter hybrid exchange functional and Lee-Yang-Parr correlation functional (B3LYP) calculations and Born-Oppenheimer molecular dynamics (BOMD) simulations have been performed to understand the stability of different anions inside fullerenes of various sizes. As expected, the stability of anion inside the fullerene depends on its size as well as on the size of the fullerene. Results show that the encapsulation of anions in larger fullerenes (smaller fullerene) is energetically favorable (not favorable). The minimum size of the fullerene required to encapsulate F(-) is equal to C(32). It is found from the results that C(60) can accommodate F(-), Cl(-), Br(-), OH(-), and CN(-). The electron density topology analysis using atoms in molecule (AIM) approach vividly delineates the interaction between fullerene and anion. Although F(-)@C(30) is energetically not favorable, the BOMD results reveal that the anion fluctuates around the center of the cage. The anion does not exhibit any tendency to escape from the cage.     

A comparative study of molecular complexation of [60]- and [70]fullerenes with 1,3,5-tribromobenzene by UV-vis spectrophotometric method

By UV-vis spectrophotometric method it has been shown that 1,3,5-tribromobenzene (TBB) forms molecular complexes of 1:2 stoichiometry with [60]- and [70]fullerenes. An isosbestic point could be detected in case of the [70]fullerene complex. The formation constant of the [60]fullerene complex is higher than that of the [70]fullerene complex at each of the four temperatures under study. This is in opposite order of the electron affinities of the two fullerenes; moreover, no charge transfer band was observed in the spectra of either complex in solution. This indicates that van der Waals forces, rather than CT interactions, are responsible for complexation. The results reveal that the C-atoms at the pentagon vertices of [60]fullerene have greater polarizing power than those in [70]fullerene.     

C2 isomers of C84F40 and C84F44 are cuboid and contain benzenoid and naphthalenoid aromatic patches

A 1:1 mixture of C84F40 and C84F44, both derived from the D2(IV) isomer, has been isolated from the fluorination of [84]fullerene with either MnF3 or CoF3 at 500 degrees C. The 1D and 2D COSY 19F NMR spectra showed that each derivative is cuboid, having benzenoid rings at four of the six octahedral sites; the two remaining sites have naphthalenoid rings for C84F40, and two slightly offset benzenoid rings for C84F44. The benzenoid rings each have six adjacent sp3-hybridised carbon atoms whilst the naphthalenoid moieties have eight, thus facilitating full delocalisation. In terms of the number and size of aromatic patches, C84F44 is the most aromatic fullerene derivative yet isolated.