Graph-theoretically planar

Any helicene may be represented by a planar graph and can therefore be referred to as graph—theoretically planar. 

Definition 2.5 A corohelicene (system) is a multiply connected, geometrically nonplanar (helicenic) polyhex, which is graph-theoretically planar. 

The requirement about graph—theoretical planarity is new. Notice that corohelicenes are not... 

Having the graph—theoretical planarity in mind, it is clear that the outer and inner perimeters of a single corohebcene always can be identified. However, an identification of the corona hole with a benzenoid may be obscured. Therefore we must rephrase the requirement that the corona hole should have a size of at least two hexagons (cf. Sect. 2.1). The following formulation is valid for corofusenes any inner perimeter should consist of at least ten edges (and vertices). 

Here the left-hand system is graph—theoretically nonplanar like the systems of Figs. 6 and 7. However, the right-hand isomer is a graph—theoretically planar system and actually a corohelicene similar to the system of Fig. 4. 

A thiepin is formally isoelectronic with the 8ic-electron 1,3,5,7-cyclooctatetraene and 1,3,5-cycloheptatrienide ion and, if planar, may actually be antiaromatic. Recently, the question of the antiaromaticity of thiepin has been the subject of interest for both synthetic and theoretical chemists. The apparent instability of the thiepin ring system is in good agreement with theoretical calculations. Dewar and Trinajstic 68) have reported that the thiepin is considered to be weakly antiaromatic (RE = — 1.45 kcal mol-1) based on PPP SCF MO calculations. On the other hand, Hess Jr. and Schaad 69) have found it to be substantially antiaromatic (RE = —0.232 J) by using the Huckel MO method. This result was also supported by a graph-theoretical treatment by Aihara 70). 

Temperley, H., Lieb, E. Relations between the percolation and colouring problem and other graph-theoretical problems associated with regular planar lattices Some exact results for the percolation problem. Proceedings of theRoyal Society of London A 322(1549), 251-280 (1971)... 

In recent years the fundamental ideas of Huckel molecular orbital theory, the Huckel rule, and other aspects of aromaticity have been extended to polyhedral three-dimensional inorganic structures regarded as aromatic like the two-dimensional aromatic hydrocarbons. Such an extension of Huckel molecular orbital theory requires recognition of its topological foundations so that they can be applied to three-dimensional structures as well as two-dimensional structures. In this connection graph theoretical methods can be used to demonstrate the close analogy between the delocalized bonding in two-dimensional planar aromatic systems such as benzene and that in three-dimensional deltahedral boranes, and carboranes. Related ideas can be shown to be applicable for metal carbonyl clusters, bare post-transition metal clusters, and polyoxometallates. ... 

Table 34. Expressions for Molecular RE, the Graph Theoretically Computed RE, REPE, and the Percentage Aromatic Character for the Conjugated Hydrocarbons Shown in Figure 2, Which Illustrate a Gradual Transition from Fully Aromatic Benzene to Fully Anti-aromatic Hypothetical (Planar) Cyclooctatetraene...

Fio. 3.7. Planar projections of molecular graphs of hydrocarbon molecules generated from theoretical charge distributions. Bond critical points are denoted by black dots. Structures 1 to 4 are normal hydrocarbons from methane to butane, 5 is isobutane, 6 is pentane, 7 is neopentane, and 8 is hexane. The remaining structures are identified in Table 3.2. The structures depicted in these diagrams are determined entirely by information contained in the electronic charge density. 

Fig. 1.16. The graph presents a comparison of theoretical and experimental vertical electron detachment energies (VDEs) for Aun in = 4-14). The optimized ground state structures (labeled A for each size) and close lying low-energy isomers are also displayed [120]. A change from planar geometry to SD-structural motifs is apparent between Aui2 and Auis ...

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