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Alternating 6-cycles in perfect matchings of graphs representing condensed benzenoid hydrocarbons. (English) Zbl 0633.05055

In this paper recurrence relations and algebraic expressions are deduced for the number of perfect matchings (Kekulé structures) and of alternating 6-cycles for all perfect matchings of graphs composed from k linearly condensed portions consisting each of \(j+1\) hexagons. These numbers are also expressed as polynomials in j, whose coefficients are rational polynomials in k which are found in an explicit form. An asymptotic ratio is obtained between the number of alternating 6-cycles in all perfect matchings and the total number of 6-cycles, as a function (40) of j.
Some applications of these results to chemistry are presented, e.g. the ‘conjugated circuits method’ which gives resonance energies of condensed benzenoid hydrocarbons, and which depends mainly on the number of perfect matchings.

MSC:

05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
92Exx Chemistry
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