Coronoid

He, W.C., and He, W.J. Peak-Valley Path Method on Benzenoid arid Coronoid Systems. 153, 195-210 (1990). 

Peak-Valley Path Method on Benzenoid and Coronoid Systems... 

Algorithmic Approaches for Deciding Whether a Benzenoid or Coronoid Hydrocarbon is Kekutean or Non-Kekul an.197... 

The present paper is a review of the P-V path method. The concept of the P-V path was proposed by Gordon and Davison in 1952. In the last few years this method has been greatly developed and has become one of the important approaches for investigating Kekule structures of benzenoid hydrocarbons. The superiority of this method is its simplicity and visualization. According to the properties of the P-V path, some algorithmic approaches have been developed for deciding whether a benzenoid or a coronoid hydrocarbon is Kekulean. In 1985, John and Sachs introduced the concept of the P-V matrix and deduced the John-Sachs theorem which states that the absolute value of the determinant of the P-V matrix of a benzenoid or a coronoid hydrocarbon G is equal to the number of Kekule structures of G. 

This article is a review of the P-V path method, which is used for investigating Kekule structures of benzenoid and coronoid (single and multiple) hydrocarbons and has been developed in the last few years. 

Only Kekulean benzenoid and coronoid hydrocarbons are known to exist. Non-Kekulean benzenoid and coronoid systems have never been synthesized [2-6], they should be polyradicals and have very low chemical stability. 

The following results are true for coronoid systems as well as benzenoids. 

Theorem 3. For a Kekulean benzenoid or coronoid system the number of peaks is equal to the number of valleys [11],... 

A = 0 is a necessary but not sufficient condition for Kekulean benzenoids and coronoids. The systems with A 4= 0 are obvious non-Kekulean systems and those non-Kekulean systems with A = 0 are called concealed non-Kekulean systems, (see Fig. 2)... 

Sachs algorithm can be used on coronoid systems, but the deletion method of convex pair can not. An example is shown in Fig. 7. (u, v) is a convex pair in the coronoid system, but in any Kekule structure of the system, the edge uv is not a double bond edge. 

In 1987, He and He used the network method to investigating P-V path structures of benzenoid and coronoid system [11],... 

Consider a benzenoid or a coronoid system G. A network N is constructed from G as follows [11, 20, 21],... 

Theorem 11 [11], A benzenoid or coronoid system G is Kekulean if and only if... 

Below we outline the proof of Theorem 16. This proof was given by He and He [12]. Lemma 1. For a benzenoid or a coronoid G with A = 0,... 

The enumeration of Kekule structures for rectangle-shaped benzenoids is treated. Combinatorial formulas for K (the Kekule structure count) are derived by several methods. The oblate rectangles, Rj(m, n), with fixed values of m are treated most extensively and used to exemplify different procedures based on the method of fragmentation (chopping, summation), a fully computerized method (fitting of polynominal coefficients), application of the John-Sachs theorem, and the transfer-matrix method. For Rj(m, n) with fixed values of n the relevant recurrence relations are accounted for, and general explicit combinatorial K formulas are reported. Finally a class of multiple coronoids, the perforated oblate rectangles, is considered in order to exemplify a perfectly explicit combinatorial K formula, an expression for arbitraty values of the parameters m and n. 

Consider a perforated oblate rectangle as depicted in Fig. 9. Like a rectangle it is a system with two parameters (m, n). A perforated rectangle, Q(m, ), belongs to the multiple coronoids it has m — 1 corona holes. 

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