Concealed 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. 

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. 

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]. 

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)... 

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 vertices of a benzenoid system can be colored by two colors, say black and white, so that first neighbors have different colors [3], Since every double bond in a Kekule structure lies between a black and a white vertex, every Kekulean benzenoid system must have equal numbers of black and white vertices. (Recall that the K = 0 benzenoids having equal numbers of black and white vertices are called concealed non-Kekulean systems [3].)... 

Below is depicted a benzenoid system 15 (with colored vertices) and two of its edge-cuts. In these edge-cuts only the vertices belonging to the fragment Fx are colored. In the first edge-cut F, has more black than white vertices. In the second edge-cut F, has more white than black vertices. This latter cut reveals that 15 is (concealed) non-Kekulean. 

A non-Kekulean benzenoid, which necessarily is pericondensed, may be obvious non-Kekulean or concealed non-Kekulean. If A > 0 for a benzenoid, then it is obvious non-Kekulean. If A = 0 and K = 0, the benzenoid is concealed non-Kekulean. 

The benzenoid systems with A = 0 coincide with the Kekuleans for h < 10. For h > 11 the concealed non-Kekuleans must be added. 

The class of concealed non-Kekuleans is defined by A = 0, K = 0. All members of this class are pericondensed. They occur at h > 11. 

Table 30 shows numbers of concealed non-Kekulean benzenoids, including the distributions into symmetry groups. Some supplements are accessible from appropriate forthcoming tables. 

Table 30. Numbers of concealed non-Kekulean benzenoids classified according to syim metry+...

Fig. 28. All concealed non-Kekulean benzenoids with h < 12, classified according to n,...

The mathematical deductions did not always come after the computer-aided analysis. In the mentioned work of Guo and Zhang [108] also the number of concealed non-Kekuleans with h = 13 (see Table 30) was reported. An analytical deduction of the same number was achieved by Jiang and Chen [109], who extended their mathematical analysis to attain at the corresponding number for h = 14. Cyvin et al. [93] offered the following comment. We wish to emphasize that these numbers (viz. 1097 and 9781) were obtained by mathematical analyses without computer aid. Brilliant achievements The former number (viz. 1097 for h = 13) was confirmed by a computer analysis of the present work. For h = 14, however, a very recent computer result of the present work deviates from the Jiang and Chen number. The new number (viz. 9804) is supposed to be correct and is therefore entered in Table 30. 

All snowflakes have vanishing color excess A = 0. Therefore they can be either normal, essentially disconnected or concealed non-Kekulean (but not obvious non-Kekulean). 

Kekulean snowflakes have been depicted or reproduced several times [71, 103, 107, 110]. The Dbh system out of these was actually identified and depicted for the first time by Hosoya [119], while three out of the four C6h systems were depicted by Cyvin et al. [120]. The 42 concealed non-Kekulean (improper) snowflakes with h = 49 have also been depicted before [107]. The 313 concealed non-Kekulean snowflakes with h = 55 have been described, and selected representatives of them have been depicted [118]. In this set there is 1 proper snowflake, which also has been depicted together with all such systems for h < 73 [103] the 7 largest of these systems (for h = 73) are reproduced elsewhere [93]. One of these papers [103] shows the computer-generated pictures of all proper snowflakes with h < 55. 

Here we give a re-edited selection of the forms of snowflakes. Figures 29, 30 and 31 display the forms of normal, essentially disconnected and concealed non-Kekulean snowflakes, respectively, both proper (D6h) and improper (C6h). Similarly, the forms of proper snowflakes in particular are displayed in Figs. 32, 33 and 34, pertaining to the normal, essentially disconnected and concealed non-Kekulean systems, respectively. 

Fig. 31. Snowflakes all concealed non-Kekulean benzenoids with hexagonal symmetry, D6h (one system) or C6h, and h < 55 5 systems with h = 43 and 42 with h = 49...

Fig. 34. Proper snowflakes all concealed non-Kekulean benzenoids with D6h symmetry and h <19...

The ten smallest concealed non-Kekulean benzenoids with regular trigonal symmetry (D3h) are depicted in Fig. 38 and taken from Cyvin et al. [110], The reader is referred to the below comments about the search for concealed non-Kekuleans with C3h symmetry. 

Fig. 38. All concealed non-Kekulean benzenoids with D3h symmetry and h < 46 3 and 7 systems with h = 40 and 43, respectively. The first system at h = 43 belongs to the class D3k (ib), all the others to Dih (ia)...

The 15 smallest concealed non-Kekuleans with D2h symmetry have been depicted [93] and are also reproduced in Fig. 39. Otherwise the forms of dihedral and... 

One of the D2h concealed non-Kekuleans with h = 14 is depicted by Gutman and Cyvin [102] under a wrong indication of its number of hexagons. 

It is clear that A = 0 holds for all Kekulean (n + e) benzenoids. Hence, if A > 0, the system is non-Kekulean (o) then it is called an obvious non-Kekulean benzenoid. But also non-Kekulean systems with A = 0 can be constructed they are called concealed non-Kekulean benzenoids. 

Concealed non-Kekulean benzenoids occur for the first time at h = 11. In Table 3 we are giving Hosoya [50] credit for the enumeration of such systems he depicted for the first time all eight of them as a group. Those for h = 12 (Table 4) were depicted by He et al. [51],... 

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