Non—Kekulean

The existence of Kekule structures in a benzenoid system is the first fundamental problem in the topological theory of benzenoid systems. It was considered as one of the most difficult open problems in this theory. Many investigations have been made in order to find necessary and sufficient conditions for the existence of Kekule structures in a benzenoid system. Some fairly simple conditions which are both necessary and sufficient have been given in the last few years. In this chapter we review the main results and give a rigorous proof for some necessary and sufficient conditions for the existence of Kekule structures in a benzenoid system. In addition, by using the above results, a construction method of some concealed non-Kekulean benzenoid systems is given. 

A Kekule structure or a 1-factor of a benzenoid system H is an independent edge set in H such that every vertex in H is incident with an edge in the edge set. A benzenoid system is said to be Kekulean if it possesses a Kekule structure, otherwise it is said to be non-Kekulean. It was first pointed out by Clar et al. [1,2] that Kekule structures are of paramount importance for the stability of benzenoid systems. In fact, up to now, no non-Kekulean benzenoid system has been synthesized by chemists. Therefore the existence of Kekule structures in a benzenoid system is a fundamental problem in the topological theory of benzenoid systems. 

It is well known that a Kekulean benzenoid system must have the same number of black and white vertices, and consequently A = 0. The benzenoid systems with A 4= 0 are said to be obvious non-Kekulean. In the following, we focus our attention on the benzenoid systems with A = = 0. 

The conditions A = 0 is necessary but not sufficient for the existence of Kekule structures. The smallest non-Kekulean benzenoid systems with A = 0 (called concealed non-Kekulean) were found during the years 1972-1986 see, e.g. Bfiinvoll et al. [16], Two of them are shown in Fig. 3. 

Some other necessary or sufficient (but not necessary and sufficient) criteria for the differentiation between Kekulean and non-Kekulean benzenoid systems can be found in a number of recently published papers [5,19-23], In particular, Sachs gave some important results [5], He introduced the concepts of a cut segment and a cut in a benzenoid system. 

From Theorem 6 and Lemma we see that there exist some concealed non-Kekulean benzenoid systems which satisfy the conditions of the lemma but not (2) of the theorem. We call them the concealed non-Kekulean benzenoid systems of type I. Such an example is shown in Fig. 6. 

It is natural to investigate the smallest concealed non-Kekulean benzenoid systems of type I. It may enable us to find some simpler necessary and sufficient conditions for benzenoid systems with small number of hexagons to have Kekule structures. 

The smallest concealed non-Kekulean benzenoid system of type I was first found by Zhang Fuji and Guo Xiaofeng [27] (see Fig. 9). 

For the benzenoid systems with h < 11, the conditions of Theorem 8 can be further simplified. As stated above, a smallest concealed non-Kekulean benzenoid system has eleven hexagons (see Fig. 3). It implies that, for benzenoid systems with h < 11, (i) of Theorem 8 is also sufficient. By Theorem 8 we can give it a simple proof. 

By computer-aided generation the authors of Ref. [16] found that there are exactly eight smallest concealed non-Kekulean benzenoid systems. In addition, He Wenchen et al. [28] asserted that there are exactly ninety-eight concealed non-Kekulean benzenoid systems with h = 12. 

Using Theorem 8, we can give a construction method for concealed non-Kekulean benzenoid systems with h < 14, which does not depend on computer-aided generation. By this construction method, we proved the above results from References [16 and 28], Furthermore, we have proved that there are exactly 1097 concealed non-Kekulean benzenoid systems with h = 13 [27, 29]. 

We would like to thank Prof. I. Gutman and S. J. Cyvin for their manuscript Kekulean and Non-Kekulean Benzenoid Hydrocarbons prior to publication [30], We would also like to acknowledge support from the NNSFC. 

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. 

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)... 

Algorithmic Approaches for Deciding Whether a Benzenoid or Coronoid Hydrocarbon is Kekulean or Non-Kekulean... 

A simple method for recognizing an obvious non-Kekulean system is by counting its peaks and valleys. 

We have the same definitions of a Kekule structure, Kekulean, non-Kekulean, double and single bonds, fixed double and fixed single bonds for generalized benzenoid systems as for benzenoid systems. 

First, we check whether H contains an equal number of peaks and valleys if not, then H is non-Kekulean in particular this happens if H consists of an isolated vertex. 

The algorithm comes to a stop when either (a) a non-Kekulean condition is reached then H is non-Kekulean, or (b) all vertices of H are deleted in this case H is Kekulean. If H is recognized as Kekulean, we recall the deleting process if we delete two end vertices of an edge, then the edge is considered as a double bond, and in this way we can constitute a Kekule structure of H. 

Theorem 4.2 demonstrates the important fact that deleting a convex pair from a benzenoid system does not alter its Kekulean/non-Kekulean character. 

Case 1. H contains a pendent vertex. We may delete a pendent vertex together with its first neighbour without changing the Kekulean/non-Kekulean nature. 

The above procedure ends when an isolated vertex is created or when all vertices of H are pairwise deleted. In the former case H is non-Kekulean in the latter case it is Kekulean, and a Kekule structure can be constructed by regarding the edge joining the deleted pair of vertices at each step as a double bond. 

Fig. 12. Exemplification of the Sheng algorithm (a) is non-Kekulean, x is an isolated vertex (b) is Kekulean, the encircled vertex pairs correspond to the double bonds in a Kekule structure...

Fig. 15. This is a non-Kekulean benzenoid system because after step 3 no paths joining p4 and v4 exist...

In this situation we can also apply the peeling algorithm, as reported in Refs. [11,12,17]. The peeling algorithm consists of the deletion of the extreme left monotonic path, followed by the deletion of pendent vertices (if any) together with their first neighbours. The procedure ends when either (a) an isolated vertex is created in this case B is non-Kekulean, or (b) all vertices are deleted then it is Kekulean. An example is given in Fig. 16. 

Fig. 16. By means of the peeling algorithm this benzenoid system is found to be non-Kekulean x is an isolated vertex...

Since all catacondensed benzenoid system are normal, we may reduce a generalized benzenoid system with fused catacondensed units without altering the Kekulean/non-Kekulean character. By inspecting Fig. 19. the reader may get a fair idea about how the reduction is made. 

Here we have reviewed some advances in rapid ways to recognize Kekulean benzenoid systems. Many important properties concerning Kekulean and non-Kekulean (generalized) benzenoid systems have been reported. 

The Sheng algorithm is to be recommended for its simplicity and rapidity. By means of this algorithm, a complicated benzenoid system can be easily reduced, and thus rapidly recognized as being Kekulean or non-Kekulean. 

A graph with an odd number of points is non-Kekulean by definition. No benzenoid hydrocarbon molecule or radical corresponding to non-Kekulean graph has ever been synthesized. The phenalene skeleton, XHIa, is the smallest non-Kekulean benzenoid. However, it is regrettably true that even phenalenyl radical is... 

As it turns out though even the simplified VB-theoretic formulations giving rise to conjugated-circuit theory or even just Kekule-structure enumeration, may become challenging for sufficiently large (perhaps formally infinite) systems, or for non-Kekulean (i.e., radicaloid) systems. It might oft be convenient if explicit enumeration of Kekule structures could be avoided. Notably for such cases there are some few alternative sorts of means by which to obtain some partial information about the system, within a VB-theoretic context. 

In the book [3] the role of both Kekule and Clar structures in various (contemporary) chemical theories as well as their relevance for practical chemistry were outlined in detail. A recent book [46] by Cyvin and the present author is devoted to the enumeration of Kekule structures of benzenoid molecules. In addition to this, the first volume of Advances in the Theory of Benzenoid Hydrocarbons contains several review articles [47-51] dealing with topics of relevance for our considerations. In order to avoid repetition and overlapping we will just briefly mention the work on the elaboration and application of the John — Sachs theorem for the enumeration of Kekule structures [52-55], the search for concealed non-Kekulean benzenoid systems [2, 56-59], examination of fully benzenoid (=all-benzenoid) systems [60-62, 135] as well as the enumeration of Kekule structures in long and random benzenoid chains [63-65]. 

The early history of the search for non-Kekulean benzenoid systems (= systems for which no Kekule structural formula can be written = systems for which K = 0) is described elsewhere (see pp. 62-66 in [3]). Some time was needed for theoretical chemists to recognize that for large benzenoids it is not quite simple to decide whether K > 0 (Kekuleans) or K = 0 (non-Kekuleans). 

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