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2.8 PERFECT MATCHINGS IN HEXAGONAL SYSTEMS The transformation from molecular structure (e.g., Fig. 2.1a) to an HS (e.g., Fig. 2.1b) leaves out the information about double valences between carbon atoms. Clearly, each carbon atom has a double valence link with exactly one of its neighboring carbon atoms. Thus, double valences correspond to a perfect matching in an HS. Therefore, an HS is the skeleton of a benzenoid hydrocarbon molecule if and only if it has a perfect matching. An HS that has at least one perfect matching is called Kekul an; otherwise, it is e called non-Kekul an. Kekul an HSs are further classi ed as either normal (if every e e edge belongs to at least one perfect matching) or essentially disconnected (otherwise). Classi cation of HSs according to the perfect matching property is summarized by Cyvin et al. [14]. An HS with a given perfect matching is called a Kekul structure in e chemistry and has great importance. Figure 2.11a and b shows two Kekul structures e that corresponds to the HS in Figure 2.1b. If the number of vertices of an HS is odd, then clearly there is no perfect matching. We denote by K(G) the number of perfect matchings of a graph G, and refer to it as the

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