Kekule graph

To use the wavefunction P of (5.1.1) one needs to evaluate its matrix elements. One way to do this is to compute representation matrices on the basis of Kekule structures, then sum the elements of these matrices. But graph-theoretic representations for the overall matrix element of may also be obtained. Using (4.2.4) with the sign convention possible for alternants, one obtains... 

A perfect matching or 1-factor of a graph G is a matching of G that covers all vertices. If, in particular, G is a BS then a 1-factor of G is usually called a Kekule structure (KS). 

Since a benzenoid system H is a bipartite graph, the existence of Kekule structures of H is equivalent to the existence of 1-factors (perfect matchings) of a bipartite graph. In 1935, P. Hall found the following necessary and sufficient conditions. 

Define the Clar transformation (C) as a simultaneous substitution of all the proper sextets by circles in a given Kekule pattern k,. followed by the suppression of the remaining double bonds into single bonds, as exemplified for graph XVI given in Fig. 1. 

Before going into these discussions let us again focus our attention on the mathematical structure of the Kekule and sextet patterns of benzenoid hydrocarbon graphs. The third theorem reads as follows ... 

The sextet rotation to the set of the Kekule patterns k, gives a directed tree graph with a root, or the root Kekule pattern, representing the hierarchical structure of kj, where each point corresponds to a Kekule pattern. 

In Figs. 3 and 4 are given the rooted directed trees derived by joining all the entries of the Kekule patterns of XVI and V with the sextet and counter-sextet rotations. One can realize how all the Kekule and sextet patterns are related to each other. Theorem 3 can be proved by showing that there is one and only one root Kekule pattern, a Kekule pattern without any proper sextet, (see Lemma) and no cyclic relation among the Kekule patterns with respect to the sextet rotation. This graph-theoretical discussion does not necessarily mean that the root Kekule pattern is the most chemically-important in the family of the Kekule patterns, but that all the patterns are mathematically related with each other. 

Figure 1 - Plots of resonance energies per site for several simple ground-state approximants. The single-Kekule-structure plot applies for general graphs, the Neel-state plot for general bipartite graphs, and the RVB plot for a special class [115] of benzenoid polymers.

The recursions of the preceding section can be alternatively cast into an especially elegant form for polymer graphs. The Kekule-structure count KL for a polymer chain of length L monomers can [139-144] quite generally be cast into the form of a trace... 

The most general Kekule-structure-count method of the present type was devised by Kasteleyn [146], though there is slightly earlier work for different special cases [33,147]. This too involves certain matrices, most simply the graph adjacency matrices /4(G) with rows columns that are labelled by the sites of G and elements that are all 0 except those Aab=+ with a b adjacent sites in G. Then Kastelyn shows how for "planar" graphs to set up a "signed" version (G) of this matrix with half of its +1 elements replaced by -1 such that... 

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