Abstract
A toroidal polyhex H(p, q, t) is a cubic bipartite graph embedded on the torus such that each face is a hexagon, which can be described by a string (p, q, t) of three integers (p≥ 1, q≥ 1, 0≤ t≤ p−1). A set \(\mathcal H\) of mutually disjoint hexagons of H(p, q, t) is called a resonant pattern if H(p, q, t) has a prefect matching M such that all haxgons in \(\mathcal H\) are M-alternating. A toroidal polyhex H(p, q, t) is k-resonant if any i (1 ≤ i ≤ k) mutually disjoint hexagons form a resonant pattern. In [16], Shiu, Lam and Zhang characterized 1, 2 and 3-resonant toroidal polyhexes H(p, q, t) for min(p, q)≥ 2. In this paper, we characterize k-resonant toroidal polyhexes H(p, 1, t). Furthermore, we show that a toroidal polyhex H(p, q, t) is k-resonant (k≥ 3) if and only if it is 3-resonant.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Altschuler A. (1973) Construction and enumeration of regular maps on the torus. Discrete Math. 4, 201–217
Chen R., Guo X. (1993) k-coverable coronoid systems. J. Math. Chem. 12, 147–162
Clar E. (1972) The Aromatic Sextet. Wiley, London
Deza M., Fowler P.W., Rassat A., Rogers K.M. (2000) Fullerenes as tilings of surfaces. J. Chem. Inf. Comput. Sci. 40, 550–558
Guo X. (2006) k-resonace in benzenoid systems, open-ended carbon nanotubes, toroidal polyhexes; and k-cycle resonant graphs. MATCH Commun. Math. Comput. Chem. 56, 439–456
Guo X., Zhang F. (2001) k-resonant benzenoid systems and k-cycle resonant graphs. J. Chem. Inf. Comput. Sci. 41(3): 480–483
E.C. Kirby, Recent work on toroidal and other exotic fullerene structures, in From Chemical Topology to Three-Dimensional Geometry, Ed. A.T. Balaban (Plenum Press, New York, 1997) pp. 263–296
Kirby E.C., Mallion R.B., Pollak P. (1993) Toridal polyhexes. J. Chem. Soc. Faraday Trans. 89(12): 1945–1953
Klein D.J. (1994) Elemental benzenoids. J. Chem. Inf. Comput. Sci. 34: 453–459
Lin K., Chen R. (1996) k-coverable polyhex graphs. Ars Combin. 43: 33–48
Marušič D. (2000) T. Pisanski, Symmetries of hexagonal molecular graphs on the torus. Croat. Chem. Acta 73(4): 969–981
Negami S. (1983) Uniqueness and faithfulness of embedding of toroidal graphs. Discrete Math. 44, 161–180
Randić M. (2003) Aromaticity of polycyclic conjugated hydrocarbons. Chem. Reviews 103(9): 3449–3605
Randić M. (1977) Aromaticity and conjugation. J. Amer. Chem. Soc. 99, 444–450
Randić M. (1976) Conjugated circuits and resonance energies of benzenoid hydrocarbons. Chem. Phys. Lett. 38, 68–70
Shiu W.C., Lam P.C.B., Zhang H. (2005) k-resonance in toroidal polyhexes. J. Math. Chem. 38(4): 451–466
Shiu W.C., Zhang H. (2006) A complete characterization for k-resonant Klein-bottle polyhexes. J. Math. Chem. doi:10.1007/s10910-006-9185-7
Thomassen C. (1991) Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface. Trans. Amer. Math. Soc. 323, 605–635
Zhang F., Chen R. (1991) When each hexagon of a hexagonal system covers it. Discrete Appl. Math. 30, 63–75
Zhang F., Wang L. (2004) k-resonance of open-ended carbon nanotubes. J. Math. Chem. 35(2): 87–103
Zhang F., Zheng M. (1992) Generalized hexagonal systems with each hexagon being resonant. Discrete Appl. Math. 36, 67–73
Zhang H., Zhang F. (1996) The Clar covering polynomial of hexagonal systems I. Discrete Appl. Math. 69, 147–167
Zhang H., Zhang F. (2000) Plane elementary bipartite graphs. Discrete Appl. Math. 105, 291–311
Zheng M. (1991) k-resonant benzenoid systems. J. Mol. Struct. (Theochem) 231, 321–334
Zheng M. (1992) Construction of 3-resonant benzenoid systems. J. Mol. Struct. (Theochem) 277: 1–14
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhang, H., Ye, D. k-resonant toroidal polyhexes. J Math Chem 44, 270–285 (2008). https://doi.org/10.1007/s10910-007-9310-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-007-9310-2