Obvious non—Kekulean

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. 

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)... 

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

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. 

A classification of the benzenoids according to the color excess (A) sorts out the obvious non-Kekuleans (A > 0) and the systems with vanishing color excess... 

The benzenoids with A > 0 are by definition the obvious non-Kekuleans, They have K = 0 and are perieondensed. Only four symmetry groups are possible for these systems, viz, D3h, C3I C2v and Cs. 

Table 29. Numbers of obvious non-Kekulean benzenoids with different colour excess (A values), classified according to symmetry...

Fig. 24. All obvious non-Kekulean benzenoids with h < 1. A values and numbers of internal vertices (nt) are indicated...

Table 29 gives a general survey of the numbers of obvious non-Kekuleans with given A values, including the distributions into symmetry groups. Extensions for the trigonal symmetries are accessible through subsequent tables. 

In Fig. 24 the forms of the obvious non-Kekuleans up to h = 7 are shown as black silhouettes. They have been given previously as dualists [55]. 

The title class refers to benzenoids which have the maximum A value, A = Amas, for a given number of hexagons (h). For A > 0 they are obvious non-Kekuleans. These systems are treated in detail by Brunvoll et al. [101]. 

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). 

Fig. 37. All (obvious) non-Kekulean benzenoids with trigonal symmetry (D3k or C3h) and h < 13 1, 2, 6, 2 and 24 systems with h = 3, 6. 9, 10 and 12, respectively the two systems with h = 10 are of the first kind, all the other of the second kind. A values are indicated...

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. 

All the non-Kekulean coronoids with h < 14 (for their numbers, see Table 1) are known to be obvious non—Kekuleans. The smallest concealed non—Kekul an (single) coronoids have h = 15, and there are exactly 23 nonisomorphic systems of this category (Cyvin SJ, BrunvoU and Cyvin 1989c Vol. 1-7.3), each of them possessing the naphthalene hole. For the subclass of single coronoids with the phenalene hole it was found that the smallest concealed non—Kekuleans have h = 16, and there are exactly 21 such systems (Cyvin SJ, BrunvoU and Cyvin 1990b). 

The Kekulean and non-Kekulean systems are those which possess Kekule structures K > 0) oi do not possess Kekule structures K = 0), respectively. In an essentially disconnected system there are fixed bonds, viz. edges which correspond to double and/or single bonds in the same positions of all Kekule structures. Kekulean systems without fixed bonds are called normal Obvious- and concealed non—Kekulean systems have A > 0 and A = 0, respectively, where A is the color excess, viz. the absolute magnitude of the difference between the numbers of peaks and valleys (1-3.2.4). Finally, the catacondensed (unbranched or branched) and the pericondensed systems have n- = 0 and n-> 0, respectively, where n- designates the number of internal vertices (I-3.3.1). 

Cyvin SJ, Gutman I (1987) Topological Properties of Benzenoid Hydrocarbons — Part XLIV — Obvious and Concealed Non-Kekulean Benzenoids. J Mol Struct (Theodiem) 150 157... 

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