Extremal benzenoid

It is easily found that the extremal benzenoids, as defined in Sect. 6.2, say A, have the formulas given by... 

The protrusive benzenoids form a subclass of the extremal benzenoids. By definition a protrusive benzenoid has a formula with no other formula in the same column above it, and no formula in the same row to the left of it in the periodic table. All pericondensed protrusive benzenoids are generated by circumscribing the extremal benzenoids. Consequently they have the formulas as given below [53],... 

The extreme-left benzenoids are defined by formulas on the staircase-like boundary so that, in each case, there is no formula in the same row to the left. Hence the extremal benzenoids without benzene form a subclass of the extreme-left benzenoids.But there exist extreme-left benzenoids, say x, which are not extremal. Their formulas are found one step up and one step to the right from every formula for the pericondensed protrusive benzenoids. Hence one obtains readily the following expression. 

The extremal benzenoids have no coves and no fjords, but this property is not valid for strictly pericondensed benzenoids in general. Figure 3 shows some counterexamples. 

Propositions 2 (a) An extremal benzenoid has no cove and no fjord. (b) An extreme-left benzenoid has no fjord. 

Suppose that C HS is the formula for an extreme-left benzenoid with a cove. By covering the cove (in the language of Section 2.2) a new benzenoid is created with one hexagon more and three more internal vertices. It has the formula C +1HS 1 which is situated in the periodic table relatively to C HS as indicated in the below scheme. At the same time the scheme shows the only possible shape of the staircase-like boundary in the vicinity of C HS and C +1Hs i, as can easily be deduced from Propositions 1 (a) and 1 (b). It is found that C HS is not compatible with an extremal benzenoid. That proves Proposition 2 (a). 

As a result of the above proof it may be stated An extreme-left benzenoid with a cove is not an extremal benzenoid. 

It was proved that there is no cove and no fjord among the extremal benzenoids. This is a necessary condition for the feature that all such benzenoids can be circumscribed. In general, the absence of coves and fjords is not a sufficient condition for this feature [8]. Nevertheless it is inferred that all extremal benzenoids, say A, can be circumscribed and not only once, but an infinite number of times. We go still a step further. Let (n s) be the formula of A and (ri s ) of drcum-A as in Eq. (11). If all the (n s) benzenoids A are circumscribed, it is inferred that the created circum-A benzenoids represent all the (ri s ) isomers. Then there is obviously the same number of the A and circum-A systems. The properties described in this paragraph are very plausible and represent the foundation of the algorithms [46, 50, 51] and general formulations [51] for constant-isomer benzenoid series, which have been put forward. Also some attempts to provide rigorous proofs are in progress. 

Let again the formulas (n s) and (ri s ) pertain to A and circum-A, respectively, where A is an extremal benzenoid. Then, by virtue of Proposition 5 (Sect. 7.1) it should be clear that circum-A must be protrusive (and hence also extremal). 

From the formula apparatus in our main reference [8], founded on the Harary-Harborth [44] relations, one obtains the following equation for the perimeter length (nj of an extremal benzenoid. 

The invariant ne, also equal to the number of external vertices, assumes its minimum given by (18) for a given h. Here the ceiling function is employed [V] is the smallest integer larger than or equal to x. Now the formula for an extremal benzenoid, A, reads in general [8]... 

An extremal coronoid (single or multiple) is defined in the same way as an extremal benzenoid (Harary and Harborth 1976 Cyvin SJ 1992c BrunvoU, Cyvin BN and Cyvin 1992b). 

This derivation is based on the assumption that the number of internal vertices in a polyhex with h hexagons, viz. n h, g), cannot be larger than it would be in an extremal benzenoid perforated by g naphthalene holes. At this stage, however, it is an open question whether such an... 

Definition S.5 A is a perfect extremal coronoid if and only if A is an extremal benzenoid. 

Consider the extremal benzenoid(s) with g hexagons. By definition they have the maximum number of internal vertices for the given say Here (Harary and... 

Inspect the extremal benzenoid(s) with a given g for the numbers of parallel edges, say P. In general there are three directions to be inspected for every system, but one or two of the directions may be equivalent by virtue of symmetry. Detect the minimum value, viz. 

In order to depict (one of) the smallest coronoid(s) with g holes (a) select an extremal benzenoid with P = and draw it so that the parallel edges are horizontal (b) convert each hexagon of this benzenoid into a naphthalene corona hole according to ... 

Consider the case of y 6. There are three extremal benzenoids with six hexagons each, viz. (I) anthanthrene, (II) benzo[ ht]perylene and (III) triangulene. For these systems one has P s 9 in most cases as illustrated below, where the horizontal edges (marked by heavy lines) are counted. 

Consider the case of = 7. Coronene is the unique extremal benzenoid with seven hexagons. Furthermore, the three orientations for counting parallel edges are all equivalent. Hence the following construction. 

Table 3.3. Values of and for extremal benzenoids with g hexagons.

Notice that this derivation does not follow the algorithm because anthracene (ChHio) is not an extremal benzenoid. It has as its number of vertices = 14 > which equals 13 (cf. 

Pj in( ) by definition always pertains to an extremal benzenoid. It follows that, for anthracene,... 

In connection with Observation 1, suppose that a coronoid C has a corona hole larger than naphthalene. Then we can imagine that a closer packing of the hexagons of C is possible by a partial filling of the corona hole so that the total number of internal vertices increases. In Observation 2 the crucial term ("perfect extremal coronoid") conforms with Definition 3.5 of Par. 3.3.4. It is reasonable to imagine that there is a critical smallest size for an extremal benzenoid, say A, so that A can be perforated with g naphthalene holes, which is taken to be the necessary condition for creating a perfect extremal tuple coronoid. 

It was stated (Par. 3.6.2) that n - = 0 is realized in a y-polyhex for every g and h. It should also be clear that, if a catacondensed y-tuple extremal coronoid exists, then it can only occur for h = h g). It is always possible, namely, to add a hexagon to a catacondensed coronoid so that it becomes pericondensed. (The same property is also found for benzenoids with one exception benzene. Accordingly, both benzene H = 1) and naphthalene H = 2) are catacondensed extremal benzenoids.) The above discussion, along with the last lines of Par. 3.5.5, lead to the following conjecture. 

For g = Aj h = 21. One obtains as a starting point the formula C78H30, which corresponds to the associated benzenoids C78H22( =29, JV=40). The actual four isomers (Cyvin SJ, BrunvoU and Cyvin 1991d) of these extremal benzenoids were inspected, but none of them could be perforated by four naphthalene holes. Consequently, we turn to h = 22, the formula C82H32 and associated benzenoids C82H24(f7=30, —40). There are 1799 isomers of these nonextremal benzenoids. In accord with Fig. 2 it is claimed that only one of them can be perforated by four naphthalene holes in a unique constellation. 

In Fig. 4 the smallest perfect extremal tuple coronoids for = 1, 2, 3, 4 and 5 are shown they occur for h = 8, 13, 19, 23 and 28, respectively. The analysis was pursued, by perforating the appropriate extremal benzenoids, in order to generate the systems with a few higher h values for each g the results are included in Fig. 4. 

Furthermore, one should observe the termination of each row in the right-hand direction. This termination results in the staircase boundary. It is determined by the formulas situated at the extreme—right of each row, namely the formulas which pertain to the extremal polyhexes, as should be dear from the definition of these systems (Par. 3.3.4). When perfect extremal p-tuple coronoids are involved, then the corresponding staircase boundary reflects a part of the staircase boundary for benzenoids. It is specifically the part which corresponds to the benzenoids associated with the tuple coronoids in question. Thus, for instance, the staircase boundary for single coronoids has the same shape as the one for benzenoids when starting from C32H14 ovalene (cf. Table 5, where the start of this staircase boundary for benzenoids is indicated by thin lines). A staircase boundary of this kind, determined by perfect extremal coronoids or extremal benzenoids, shall be referred to as a perfect staircase (boundary). If imperfect extremal coronoids are involved we shall call it an imperfect staircase. 

A perfect extremal coronoid or an extremal benzenoid is characterized by h, n ) in the notation of eqns. (58) and (62). Introduce... 

The systems benzene, coronene, drcumcoronene, didrcumcoronene,. constitute a class of extremal benzenoids. Their numbers of hexagons Ef) and numbers of internal vertices are given by (Brunvoll and Cyvin SJ 1990)... 

Properties. Many properties of extremal benzenoids, A, can immediately be adapted to extremal single coronoids, A. Relevant properties of A are treated in several places (Cyvin SJ, Cyvin, BrunvoU, Gutman and John 1993 Cyvin SJ, Cyvin and BrunvoU 1993d 1993e) with more or less rigorous proofs therein. Here we shall not conduct special proofs for the adaptations of the different properties to the A coronoids. We are particularly interested in properties of A which pertain to formations (or absence of formations) on the perimeter of A. By the adaptation of these properties to A one correlates the outer perimeter of an A system with the (only) perimeter of an A. 

An extreme coronoid (not to be confused with an extremal coronoid) is analogous with an extreme benzenoid (with a few exceptions specified below) cf. Cyvin SJ (1992c), which has also been called an extreme—left benzenoid (Cyvin SJ 1992c Cyvin SJ, Cyvin and Brunvoll 1993e). 

---