Extreme—left benzenoid

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 smallest extreme-left benzenoids which are not extremal, have the formula C25H13 (h = 7) cf. Table 5. 

The pericondensed extreme-left benzenoids constitute a subclass of the strictly pericondensed benzenoids in the sense of Dias [12, 15, 19, 21, 55-57] they are defined by having all their internal vertices connected and no catacondensed appendages. Phenalene, C13H9, which has only one internal vertex, is reckoned among the strictly pericondensed benzenoids. An equivalent definition in a most succint form reads ... 

The formulas at the extreme left in the periodic table for benzenoid hydrocarbons, except C10H8 (naphthalene), represent exclusively strictly pericondensed benzenoids [12, 16, 21], viz. the pericondensed extreme-left benzenoids. Formulas for ds< 0 and not at the extreme left represent both strictly pericondensed and non-strictly pericondensed benzenoids. For ds > 0 there are no strictly pericondensed benzenoids [56]. 

The table starts with the number of h = 15 catacondensed benzenoids, which has not been published previously (but see the preceding chapter). In general most of the numbers, especially for the higher h values, pertain to extreme-left benzenoids. This is the case for all entries at h > 39. 

The fundamental aufbau principle is illustrated in Fig. 4. Here the aufbau units which come into operation are exactly the elementary aufbau units. The corresponding attachments of (i) C4H2, (ii) C3H and (iii) C2 are illustrated in Fig. 5 and pertain to the first example in each row (i)—(iii) of Fig. 4. But notice that the three types (a)-(c) specified in the Principles above correspond to the additions (ii), (i) and (iii) in that order. The formulation was chosen to be that way for several reasons, mainly because the type (a) is often sufficient for the aufbau, and because of our concern about extreme-left benzenoids [8] (see below). The last paragraph of the Principles gives rise to some general rules as specified in the following. 

The extreme-left benzenoids have by definition C HS formulas at the extreme left of the rows of the periodic table [8]. Since these formulas have no formula to the left in the periodic table all the pericondensed extreme-left benzenoids (i.e. for h > 2) can be generated by the two schemes (a) and (c). 

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. 

In order to prove Proposition 2 (b) suppose that C HS is the formula for an extreme-left benzenoid with a fjord. By immersing a hexagon in the fjord a new benzenoid is created with four more internal vertices. The position of the new formula, viz. C 2HS, is indicated below. Now it is impossible to construct a... 

Proposition 4 An extreme-left benzenoid in the row just above the top of a three-formula step has no cove. 

Since each G is an extreme-left benzenoid (and even extremal) only the two schemes (a) and (c ) of the special aufbau procedure (cf. Sect. 11) are of interest here. The cardinality of C HV (for h > 2) can accordingly be split into two parts as is symbolized by ... 

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). 

A stronger statement than Lemma 3.2 is that a benzenoid system contains at least two free edges in each of the three edge directions. In fact the vertical, extreme left (and right) edge is a free edge (see Fig. 5.). 

Benzenoid (chemical) isomers are, in a strict sense, the benzenoid systems compatible with a formula C HS. Several invariants, including the Dias parameter, are treated and relations between them are given. Many of the relations involve upper and lower bounds. The periodic table for benzenoid hydrocarbons is revisited and new aspects of it are pointed out. In this connection some new classes of benzenoids are defined extreme-left, protrusive and circular. Extensive tables of enumeration data for benzenoid isomers are presented. Some of their forms are displayed in figures. 

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],... 

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