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.     

Leapfrog fullerenes have many perfect matchings

A leapfrog fullerene on p vertices contains at least 2 ^p/8 different perfect matchings.     

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.     

Some aspects of the symmetry and topology of possible carbon allotrope structures

Elemental carbon has recently been shown to form molecular polyhedral allotropes known as fullerenes in addition to the familiar graphite and diamond known since antiquity. Such fullerenes contain polyhedral carbon cages in which all vertices have degree 3 and all faces are either pentagons or hexagons. All known fullerenes are found to satisfy the isolated pentagon rule (IPR) in which all pentagonal faces are completely surrounded by hexagons so that no two pentagonal faces share an edge. The smallest fullerene structures satisfying the IPR are the known truncated icosahedral C₆₀ of I _h symmetry and ellipsoidal C₇₀ of D _5h symmetry. The multiple IPR isomers of families of larger fullerenes such as C₇₆, C₇₈, C₈₂ and C₈₄ can be classified into families related by the so-called pyracylene transformation based on the motion of two carbon atoms in a pyracylene unit containing two linked pentagons separated by two hexagons. Larger fullerenes with 3ν vertices can be generated from smaller fullerenes with ν vertices through a so‐called leapfrog transformation consisting of omnicapping followed by dualization. The energy levels of the bonding molecular orbitals of fullerenes having icosahedral symmetry and 60n ² carbon atoms can be approximated by spherical harmonics. If fullerenes are regarded as constructed from carbon networks of positive curvature, the corresponding carbon allotropes constructed from carbon networks of negative curvature are the polymeric schwarzites. The negative curvature in schwarzites is introduced through heptagons or octagons of carbon atoms and the schwarzites are constructed by placing such carbon networks on minimal surfaces with negative Gaussian curvature, particularly the so-called P and D surfaces with local cubic symmetry. The smallest unit cell of a viable schwarzite structure having only hexagons and heptagons contains 168 carbon atoms and is constructed by applying a leapfrog transformation to a genus 3 figure containing 24 heptagons and 56 vertices described by the German mathematician Klein in the 19th century analogous to the construction of the C₆₀ fullerene truncated icosahedron by applying a leapfrog transformation to the regular dodecahedron. Although this C₁₆₈ schwarzite unit cell has local O _h point group symmetry based on the cubic lattice of the D or P surface, its larger permutational symmetry group is the PSL(2,7) group of order 168 analogous to the icosahedral pure rotation group, I, of order 60 of the C₆₀ fullerene considered as the isomorphous PSL(2,5) group. The schwarzites, which are still unknown experimentally, are predicted to be unusually low density forms of elemental carbon because of the pores generated by the infinite periodicity in three dimensions of the underlying minimal surfaces. This revised version was published online in July 2006 with corrections to the Cover Date.     

Nonclassical fullerenes with a heptagon violating the pentagon adjacency penalty rule

Nonclassical fullerenes with heptagon(s) and their derivatives have attracted increasing attention, and the studies on them are performing to enrich the chemistry of carbon. Density functional theory calculations are performed on nonclassical fullerenes C_n (n = 46, 48, 50, and 52) to give insight into their structures and stability. The calculated results demonstrate that the classical isomers generally satisfy the pentagon adjacency penalty rule. However, the nonclassical isomers with a heptagon are more energetically favorable than the classical ones with the same number of pentagon–pentagon bonds (B₅₅ bonds), and many of them are even more stable than some classical isomers with fewer B₅₅ bonds. The nonclassical isomers with the lowest energy are higher in energy than the classical ones with the lowest energy, because they have more B₅₅ bonds. Generally, the HOMO–LUMO gaps of the former are larger than those of the latter. The sphericity and asphericity are unable to rationalize the unique stability of the nonclassical fullerenes with a heptagon. The pyramidization angles of the vertices shared by two pentagons and one heptagon are smaller than those of the vertices shared by two pentagons and one hexagon. It is concluded that the strain in the fused pentagons can be released by the adjacent heptagons partly, and consequently, it is a common phenomenon for nonclassical fullerenes to violate the pentagon adjacent penalty rule. These findings are heuristic and conducive to search energetically favorable isomers of C_n, especially as n is 62, 64, 66, and 68, respectively. © 2010 Wiley Periodicals, Inc. J Comput Chem, 2010     

Finding more perfect matchings in leapfrog fullerenes

We use some recent results on the existence of long cycles in leapfrog fullerenes to establish new exponential lower bounds on the number of perfect matchings in such graphs. The new bounds are expressed in terms of Fibonacci numbers.     

A comparison between 1-factor count and resonant pattern count in plane non-bipartite graphs

The concept of resonant (or Clar) pattern is extended to a plane non-bipartite graph G in this paper: a set of disjoint interior faces of G is called a resonant pattern if such face boundaries are all M-conjugated cycles for some 1-factor (Kekulé structure or perfect matching) M of G. In particular, a resonant pattern of benzenoids and fullerenes coincides with a sextet pattern. By applying a novel approach, the principle of inclusion and exclusion in combinatorics, we show that for any plane graphs, 1-factor count is not less than the resonant pattern count, which generalize the corresponding results in benzenoid systems and plane bipartite graphs. Applications to fullerenes are also discussed.AMS Subject classification: 05C70, 05C90, 92E10     

Fullerene graphs have exponentially many perfect matchings

A fullerene graph is a planar cubic 3-connected graph with only pentagonal and hexagonal faces. We show that fullerene graphs have exponentially many perfect matchings.     

Perfect Matchings in Bipartite Lattice Animals: Lower Bounds and Realizability

In the first part of this paper we establish sharp lower bounds on the number of perfect matchings in benzenoid graphs and polyominoes. The results are then used to determine which integers can appear as the number of perfect matchings of infinitely many benzenoids and/or polyominoes. Finally, we consider the problem of concealed non-Kekuléan polyominoes. It is shown that the smallest such polyomino has 15 squares, and that such polyominoes on n squares exist for all n ≥ 15.     

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.     

Perfect matchings in random pentagonal chains

Let G be a (molecule) graph. A perfect matching, or kekulé structure and dimer covering, in a graph G is a set of pairwise nonadjacent edges of G that spans the vertices of G. In this paper, we obtained the explicit expression for the expectation of the number of perfect matchings in random pentagonal chains. Our result shows that, for any polygonal chain \(Q_{n}\) with odd polygons, the number of perfect matchings can be determined by their concatenation LA-sequence.     

A lower bound on the number of elementary components of essentially disconnected generalized polyomino graphs

An essentially disconnected generalized polyomino graph is defined as a generalized polyomino graph with some perfect matchings and forbidden edges. The number of perfect matchings of a generalized polyomino graph G is the product of the number of perfect matchings of each elementary component in G. In this paper, we obtain a lower bound on the number of elementary components of essentially disconnected generalized polyomino graphs.     

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.     

New structural parameters of fullerenes for principal component analysis

The Kekulé structure count and the permanent of the adjacency matrix of fullerenes are related to structural parameters involving the presence of contiguous pentagons p, q, r, q/p and r/p, where p is the number of edges common to two pentagons, q is the number of vertices common to three pentagons and r is the number of pairs of nonadjacent pentagons adjacent to another common pentagon. The cluster analysis of the structural parameters allows classification these parameters. Principal component analysis (PCA) of the structural parameters and the cluster analyses of the fullerenes permit their classification. PCA clearly distinguishes five classes of fullerenes. The cluster analysis of fullerenes is in agreement with PCA classification. Cluster analysis shows greatest similarity for the q–q/p and r–r/p pairs. PCA provides five orthogonal factors F ₁–F ₅. The use of F ₁ gives an error of 28%. The inclusion of F ₂ decreases the error to 2%.From the Proceedings of the 28th Congreso de Químicos Teóricos de Expresión Latina (QUITEL 2002)     

On the formation of smaller p‐block endohedral fullerenes: Bonding analysis in the E@C20 (E = Si,Ge, Sn,Pb) series from relativistic DFT calculations

Experimentally characterized endohedral metallofullerenes are of current interest in expanding the range of viable fullerenic structures and their applications. Smaller metallofullerenes, such as M@C₂₈, show that several d‐ and f‐block elements can be efficiently confined in relatively small carbon cages. This article explores the potential capabilities of the smallest fullerene cage, that is, C₂₀, to encapsulate p‐block elements from group 14, that is, E = Si, Ge, Sn, and Pb. Our interest relates to the bonding features and optical properties related to E@C₂₀. The results indicate both s‐ and p‐type concentric bonds, in contrast to the well explored endohedral structures encapsulating f‐block elements. Our results suggest the E@C₂₀ series to be a new family of viable endohedral fullerenes. In addition spectroscopic properties related to electron affinity, optical, and vibrational were modeled to gain further information useful for characterization. Characteristic optical patterns were studied predicting a distinctive first peak located between 400 and 250 nm, which is red‐shifted going to the heavier encapsulated Group 14 atoms. Electron affinity properties expose different patterns useful to differentiate the hollow C₂₀ fullerene to the proposed p‐block endohedral counterparts. © 2017 Wiley Periodicals, Inc.     

On Some Structural Properties of Fullerene Graphs

We show how some important structural properties of general fullerene graphs follow from the recently proved fact that all fullerene graphs are cyclically 4-edge connected. These properties, in turn, give us upper and lower bounds for various graph invariants. In particular, we establish the best currently known lower bound for the number of perfect matchings in fullerene graphs.     

Electrochemically formed fullerene-based polymeric films

The recent results of investigations involving the electrochemical formation of polymers containing fullerenes and studies of their properties and applications are critically reviewed. From a structural point of view, these polymers can be divided into four main categories including (1) polymers with fullerenes physically incorporated into the foreign polymeric network without forming covalent bonds, (2) fullerene homopolymers formed via [2+2] cycloaddition, (3) “pearl necklace” polymers with fullerenes mutually linked covalently to form polymer chains, and (4) “charm bracelet” polymers containing pendant fullerene substituents. The methods of electrochemical polymerization of these systems are described and assessed. The structural features and properties of the electrochemically prepared polymers and their chemically synthesized analogs are compared. Polymer films containing fullerenes are electroactive in the negative potential range due to electroreduction of the fullerene moieties. Related films made with fullerenes derivatized with electron-donating moieties as building blocks are electroactive in both the negative and positive potential range. These can be regarded as “double cables” as they exhibit both p- and n-doping properties. Fullerene-based polymers may find numerous applications. For instance, they can be used as charge-storage and energy-converting materials for batteries and photoactive units of photovoltaic cell devices, respectively. They can be also used as substrates for electrochemical sensors and biosensors. Films of the C₆₀/Pt and C₆₀/Pd polymers containing metallic nano-particles of platinum and palladium, respectively, effectively catalyze the hydrogenation of olefins and acetylenes. Laser ablation of electrochemically formed C₆₀/M and C₇₀/M polymer films (M=Pt or Ir) results in fragmentation of the fullerenes leading to the formation of hetero-fullerenes, such as [C₅₉M]⁺ and [C₆₉M]⁺.Dedicated to Professor Dr. Alan M. Bond on the occasion of his 60th birthday.     

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of have been discussed. MSC 2000: 05C17, 05C35     

On the anti-Kekulé number of leapfrog fullerenes

The Anti-Kekulé number of a connected graph G is the smallest number of edges that have to be removed from G in such way that G remains connected but it has no Kekulé structures. In this paper it is proved that the Anti-Kekulé number of all fullerenes is either 3 or 4 and that for each leapfrog fullerene the Anti-Kekulé number can be established by observing finite number of cases not depending on the size of the fullerene.     

2‐Aminoethanol Extraction as a Method for Purifying Sc3N@C80 and for Differentiating Classes of Endohedral Fullerenes on the Basis of Reactivity

Extraction with 2‐aminoethanol is an inexpensive method for removing empty cage fullerenes from the soluble extract from electric‐arc‐generated fullerene soot that contains endohedral metallofullerenes of the type Sc₃N@C_2n (n=34, 39, 40). Our method of separation exploits the fact that C₆₀, C₇₀, and other larger, empty cage fullerenes are more susceptible to nucleophilic attack than endohedral fullerenes and that these adducts can be readily extracted into 2‐aminoethanol. This methodology has also been employed to examine the reactivity of the mixture of soluble endohedral fullerenes that result from doping graphite rods used in the Krätschmer–Huffman electric‐arc generator with the oxides of Y, Lu, Dy, Tb, and Gd. For example, with Y₂O₃, we were able to detect by mass spectrometry several new families of endohedral fullerenes, namely Y₃C₁₀₈ to Y₃C₁₂₆, Y₃C₁₀₇ to Y₃C₁₂₅, Y₄C₁₂₈ to Y₄C₁₄₆, that resisted reactivity with 2‐aminoethanol more than the empty cage fullerenes and the mono‐ and dimetallo fullerenes. The discovery of the family Y₃C₁₀₇ to Y₃C₁₂₅ with odd numbers of carbon atoms is remarkable, since fullerene cages must involve even numbers of carbon atoms. The newly discovered families of endohedral fullerenes with the composition M₄C_2n (M=Y, Lu, Dy, Tb, and Gd) are unusually resistant to reaction with 2‐aminoethanol. Additionally, the individual endohedrals, Y₃C₁₁₂ and M₃C₁₀₂ (M=Lu, Dy, Tb and Gd), were remarkably less reactive toward 2‐aminoethanol.