Fullerenes with the maximum Clar number

The Clar number of a fullerene is the maximum number of mutually resonant disjoint hexagons in the fullerene. It is known that the Clar number of a fullerene with n vertices is bounded above by ? n / 6 ? - 2 , where ? x ? represents the largest integer not greater than x . We show that there are no fullerenes with n ? 2 ( mod 6 ) vertices attaining this bound. In other words, the Clar number for a fullerene with n ? 2 ( mod 6 ) vertices is bounded above by ? n / 6 ? - 3 . Moreover, we show that two experimentally produced fullerenes C80:1( D 5 d ) and C80:2( D 2 ) attain the bound ? n / 6 ? - 3 . Finally, we present a graph-theoretical characterization for fullerenes, whose order n is congruent to 2 (respectively, 4) modulo 6, achieving the maximum Clar number ? n / 6 ? - 3 (respectively, ? n / 6 ? - 2 ).

[1]  D. Manolopoulos,et al.  An Atlas of Fullerenes , 1995 .

[2]  E. Clar.  The aromatic sextet , 1972 .

[3]  H. Abeledo,et al.  Unimodularity of the Clar number problem , 2007 .

[4]  Kenneth Appel,et al.  The Four-Color Problem , 1978 .

[5]  R. King,et al.  Applications of Graph Theory and Topology in Inorganic Cluster and Coordination Chemistry , 1993 .

[6]  H. Kroto,et al.  13 C NMR spectroscopy of C76, C78, C84 and mixtures of C86–C102; anomalous chromatographic behaviour of C82, and evidence for C70H12 , 1993 .

[7]  Kathryn Fraughnaugh,et al.  Introduction to graph theory , 1973, Mathematical Gazette.

[8]  R. Whetten,et al.  Fullerene Isomerism: Isolation of C2v,-C78 and D3-C78 , 1991, Science.

[9]  Patrick W. Fowler,et al.  Boron–nitrogen analogues of the fullerenes: the isolated-square rule , 1996 .

[10]  T. L. Saaty,et al.  The four-color problem: Assaults and conquest , 1977 .

[11]  Dong Ye,et al.  A combination of Clar number and Kekulé count as an indicator of relative stability of fullerene isomers of C60 , 2010 .

[12]  Xiaofeng Wang,et al.  Packing resonant hexagons in fullerenes , 2014, Discret. Optim..

[13]  P. Fowler,et al.  Faraday communications. The higher fullerences: a candidate for the structure of C78 , 1991 .

[14]  K. Turksen,et al.  Isolation and characterization , 2006 .

[15]  Heping Zhang,et al.  New Lower Bound on the Number of Perfect Matchings in Fullerene Graphs , 2001 .

[16]  Heping Zhang,et al.  A note on the cyclical edge-connectivity of fullerene graphs , 2008 .

[17]  Heping Zhang,et al.  Forcing matching numbers of fullerene graphs , 2010, Discret. Appl. Math..

[18]  Riste Škrekovski,et al.  Cyclic edge-cuts in fullerene graphs , 2008 .

[19]  Riste Škrekovski,et al.  Cyclic 7-edge-cuts in fullerene graphs , 2008 .

[20]  Heping Zhang,et al.  Extremal fullerene graphs with the maximum Clar number , 2009, Discret. Appl. Math..

[21]  James Avery,et al.  Program Fullerene: A software package for constructing and analyzing structures of regular fullerenes , 2013, Journal of computational chemistry.

[22]  Harold W. Kroto,et al.  Isolation, separation and characterisation of the fullerenes C60 and C70 : the third form of carbon , 1990 .

[23]  Dragan Marusic.  Hamilton Cycles and Paths in Fullerenes , 2007, J. Chem. Inf. Model..

[24]  Heping Zhang,et al.  The Clar Covering Polynomial of Hexagonal Systems I , 1996, Discret. Appl. Math..

[25]  Patrick W. Fowler,et al.  Faraday communications. An end to the search for the ground state of C84 , 1992 .

[26]  Rui Yang,et al.  2-resonant Fullerenes , 2012, Eur. J. Comb..

[27]  Heping Zhang,et al.  An Upper Bound for the Clar Number of Fullerene Graphs , 2007 .

[28]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[29]  Isao Ikemoto,et al.  NMR characterization of isomers of C78, C82 and C84 fullerenes , 1992, Nature.

[30]  Branko Grünbaum,et al.  The Number of Hexagons and the Simplicity of Geodesics on Certain Polyhedra , 1963, Canadian Journal of Mathematics.

[31]  S. C. O'brien,et al.  C60: Buckminsterfullerene , 1985, Nature.

[32]  Patrick W. Fowler,et al.  Localised models and leapfrog structures of fullerenes , 1992 .

[33]  M. Randic.  Aromaticity of polycyclic conjugated hydrocarbons. , 2003, Chemical reviews.

[34]  Patrick W. Fowler,et al.  Leapfrog transformations and polyhedra of Clar type , 1994 .

[35]  Tomislav Došlić,et al.  Cyclical Edge-Connectivity of Fullerene Graphs and (k, 6)-Cages , 2003 .

[36]  Pierre Hansen,et al.  The Clar number of a benzenoid hydrocarbon and linear programming , 1994 .

[37]  Patrick W. Fowler,et al.  The leapfrog principle: a rule for electron counts of carbon clusters , 1987 .

[38]  Chunru Wang,et al.  Production and isolation of an ellipsoidal C80 fullerene , 2000 .

[39]  R. Whetten,et al.  Isolation of C76, a chiral (D 2) allotrope of carbon , 1991, Nature.

[40]  Elizabeth J. Hartung.  Fullerenes with complete Clar structure , 2013, Discret. Appl. Math..