k-resonance in Toroidal Polyhexes*

This paper considers the k -resonance of a toroidal polyhex (or toroidal graphitoid) with a string (p, q, t) of three integers (p ≥ 2, q ≥ 2, 0 ≤ t ≤ p − 1). A toroidal polyhex G is said to be k-resonant if, for 1≤ i ≤ k, any i disjoint hexagons are mutually resonant, that is, G has a Kekulé structure (perfect matching) M such that these hexagons are M-alternating (in and off M). Characterizations for 1, 2 and 3-resonant toroidal polyhexes are given respectively in this paper.

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

[2]  Haruo Hosoya,et al.  Sextet polynomial. A new enumeration and proof technique for the resonance theory applied to the aromatic hydrocarbons. , 1975 .

[3]  M. Randic.  Conjugated circuits and resonance energies of benzenoid hydrocarbons , 1976 .

[4]  M. Randic.  Aromaticity and conjugation , 1977 .

[5]  Seiya Negami,et al.  Uniqueness and faithfulness of embedding of toroidal graphs , 1983, Discret. Math..

[6]  N. Biggs.  MATCHING THEORY (Annals of Discrete Mathematics 29) , 1988 .

[7]  Douglas J. Klein,et al.  Elemental carbon cages , 1988 .

[8]  Maolin Zheng,et al.  The K-resonant benzenoid systems , 1991 .

[9]  Carsten Thomassen,et al.  Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface , 1991 .

[10]  Fuji Zhang,et al.  When each hexagon of a hexagonal system covers it , 1991, Discret. Appl. Math..

[11]  Fuji Zhang,et al.  Generalized hexagonal systems with each hexagon being resonant , 1992, Discret. Appl. Math..

[12]  Zhang Fuji,et al.  Generalized hexagonal systems with each hexagon being resonant , 1992 .

[13]  Maolin Zheng,et al.  Construction of 3-resonant benzenoid systems , 1992 .

[14]  Rong-si Chen,et al.  k-coverable coronoid systems , 1993 .

[15]  Douglas J. Klein.  Elemental Benzenoids , 1994, J. Chem. Inf. Comput. Sci..

[16]  Rong-si Chen,et al.  K-coverable polyhex graphs , 1996, Ars Comb..

[17]  Douglas J. Klein,et al.  Resonance in Elemental Benzenoids , 1996, Discret. Appl. Math..

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

[19]  H. Dai,et al.  Fullerene 'crop circles' , 1997, Nature.

[20]  Edward C. Kirby,et al.  How To Enumerate the Connectional Isomers of a Toroidal Polyhex Fullerene , 1998, J. Chem. Inf. Comput. Sci..

[21]  Edward C. Kirby,et al.  How To Enumerate the Connectional Isomers of a Toroidal Polyhex Fullerene , 1998, J. Chem. Inf. Comput. Sci..

[22]  Peter E. John,et al.  Kekulé Count in Toroidal Hexagonal Carbon Cages , 1998 .

[23]  Gordon G. Cash.  A Simple Means of Computing the Kekulé Structure Count for Toroidal Polyhex Fullerenes , 1998, J. Chem. Inf. Comput. Sci..

[24]  Patrick W. Fowler,et al.  Fullerenes as Tilings of Surfaces , 2000, J. Chem. Inf. Comput. Sci..

[25]  D. Maru,et al.  Symmetries of Hexagonal Molecular Graphs on the Torus , 2000 .

[26]  Heping Zhang,et al.  Plane elementary bipartite graphs , 2000, Discret. Appl. Math..

[27]  Fuji Zhang,et al.  k-Resonant Benzenoid Systems and k-Cycle Resonant Graphs , 2001, J. Chem. Inf. Comput. Sci..

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

[29]  Lusheng Wang,et al.  k-Resonance of Open-Ended Carbon Nanotubes , 2004 .