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3486 Chemical Reviews, 2003, Vol. 103, No. 9 Randic´ Figure 40. Benzenoid polymer. types to connecting CC bonds (top of Figure 39), the K for the monomer unit is that of a phenanthrene fragment with K ) 5, but with DS,SD assignment the monomer unit becomes a diphenylene fragment with K ) 4, etc. In this way we obtain the transfer matrix shown at the bottom of Figure 39. For any finite molecule which is a member of a family that is based on the same repeating monomer unit, the number of Kekule´ valence structures is given by the formula KN ) A(λ1)N + B(λ2)N Figure 39.

The number of Kekule´ structures for the molecule is given by K ) K1K2 + k1k2, which for the benzenoid shown in Figure 30 gives 170 + 35 ) 205. If all rings of a fragment are erased or crossed, then the corresponding ki ) 1. For families of cata-condensed benzenoids in which every benzene ring is a “kink” ring (illustrated in Figure 31), the GD algorithm immediately yields Fibonacci numbers 2, 3, 5, 8, 13, 21, 34, ... as the number of Kekule´ valence structures for the corresponding cata-condensed benzenoids.

The last row of Figure 47 shows all conceivable R3 conjugated circuits involving ring A (defined by alternating CC single and CC double bonds on the periphery of phenanthrene, anthracene, Aromaticity of Polycyclic Conjugated Hydrocarbons Chemical Reviews, 2003, Vol. 103, No. 9 3489 Figure 46. Enumeration of linearly independent conjugated circuits for 10 symmetry-non-equivalent Kekule´ valence structures of benzo[ghi]perylene by inscribing integers that indicate the size of conjugated circuits involving individual rings.

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