Croatica Chemica Acta, Vol. 90 No. 3, 2017.
Original scientific paper
https://doi.org/10.5562/cca3212
Molecular Complexity of Certain Homologous Series of Condensed Benzenoid Hydrocarbons: Limiting Values of the Patency Index and the Index of Spanning-Tree Density
Roger B. Mallion
; School of Physical Sciences, University of Kent, Canterbury CT2 7NH, England, United Kingdom
Paul Pollak
; The King’s School, Canterbury CT1 2ES, England, United Kingdom
Paweł J. Skrzyński
; The Canterbury High School, Canterbury CT2 8QA, England, United Kingdom
Abstract
Two indices of molecular complexity, the Patency Index (2017) and the Spanning-Tree Density (2003), are applied to three homologous series of condensed benzenoid hydrocarbons. Calculation of the Spanning-Tree Density requires finding the number of spanning trees in a given molecular graph, which may be achieved by applying the Cycle Theorem (2004) or, in the case of planar graphs in a planar embedding, the theorem of Gutman, Mallion & Essam (1983). To compute the Patency Index, it is necessary to count the number of ladders in the embedded molecular graph. This is done by means of the Dual Cycle Theorem (2017). In the latter, a ladder is conceived of as the edge-set relating to faces as the edge-set of a spanning tree relates to vertices. For a planar graph in a planar embedding, the number of ladders is equal to the number of spanning trees. The three homologous series investigated here are the Linear [n]-Acenes (An), the [n]-Phenacenes (Phn) and the [n]-Helicenes (Hn) (the latter of which are geometrically non-planar but graph-theoretically planar). For these three series, the Spanning-Tree Densities and the Patency Indices may be obtained in closed form and so their behaviour as n → ∞ is easily examined. Though neither index distinguishes between individual members of the three series (An, Phn and Hn) for a specified value of n, this does not preclude the possibility that, within each series, either index may exhibit correlation with physical or chemical data.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Keywords
molecular complexity; spanning trees; ladders of embedded graphs; patency index; spanning-tree density
Hrčak ID:
190804
URI
Publication date:
18.12.2017.
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