Balaban, Alexandru T.; Matematični kolokvij september 2007

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Applications of discrete mathematics to benzenoid hydrocarbons

Alexandru T. Balaban

Texas A&M University at Galveston, USA

13. september 2007

Benzenoid hydrocarbons (benzenoids) are prototypes of aromatic compounds and as constituents of petroleum they are important starting materials of plastics, pesticides, and medicinal drugs. Some of them are also unwanted contaminants in the environment, e.g. the carcinogenic benzopyrene present in exhaust gases. Historically, the prominence of German chemistry with its applications in dyes and pharmaceuticals started in the second half of the 19th century after Kekulé explained the structure of benzene (CH)6 by its cyclic conjugated formula. The high stability of benzenoids is due to a special type of delocalization of their π-electrons (aromaticity, a term that is no longer connected with their odors but with the number of delocalized π-electrons). Benzenoid rings sharing a CC bond are said to be condensed. Benzenoids that have Kekulé structures (1-factors, or dimer coverings) are stable, those that do not are unstable free radicals with unpaired electrons. The inner duals (dualists) allow the classification of benzenoids into cata-condensed systems (catafusenes) when their dualists are acyclic, perifusenes when they have 3-membered rings, and coronoids when they have larger rings. A notation for dualists based on digits 0, 1, and 2 allows finding isoarithmic benzenoids which have exactly the same number of Kekulé structures. Clar structures of benzenoids indicate sextet rings that have three double bonds, and the stability of isomeric kekuléan benzenoids is higher for structures that have a higher number of sextet rings. A few newer applications of discrete mathematics to benzenoids will be presented:

  • Benzenoids with the highest numbers of Kekulé structures;
  • Benzenoids that have either sextet rings or "empty" rings;
  • Partition of π-electrons among benzenoid rings;
  • Signature of benzenoids.

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