Graphs to chemical structures. 4: Combinatorial enumeration of planted three-dimensional trees as stereochemical models of monosubstituted alkanes |
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Authors: | Shinsaku Fujita |
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Institution: | (1) Department of Chemistry and Materials Technology, Kyoto Institute of Technology, Matsugasaki, Sakyoku, Kyoto 606-8585, Japan |
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Abstract: | Planted three-dimensional (3D) trees, which are defined as a 3D version of planted trees, are enumerated by means of Fujita’s
proligand method formulated in Parts 1–3 of this series Fujita in Theor Chem Acc 113:73–79, 80–86, 2005; Fujita in Theor
Chem Acc 115:37–53, 2006]. By starting from the concepts of proligand and promolecule introduced previously Fujita in Tetrahedron
47:31–46, 1991], a planted promolecule is defined as a 3D object in which the substitution positions of a given 3D skeleton are occupied by a root and proligands.
Then, such planted promolecules are introduced as models of planted 3D-trees. Because each of the proligands in a given planted
promolecule is regarded as another intermediate planted promolecule in a nested fashion, the given planted promolecule is
recursively constructed by a set of such intermediates planted promolecules. The recursive nature of such intermediate planted
promolecules is used to derive generating functions for enumerating planted promolecules or planted 3D-trees. The generating
functions are based on cycle indices with chirality fittingness (CI-CFs), which are composed of three kinds of sphericity
indices (SIs), i.e., a
d
for homospheric cycles, c
d
for enantiospheric cycles, and b
d
for hemispheric cycles. For the purpose of evaluating c
d
recursively, the concept of diploid is proposed, where the nested nature of c
d
is demonstrated clearly. The SIs are applied to derive functional equations for recursive calculations, i.e., a(x), c(x
2), and b(x). Thereby, planted 3D-trees or equivalently monosubstituted alkanes as stereoisomers are enumerated recursively by counting
planted promolecules. The resulting values are collected up to 20 carbon content in a tabular form. Now, the enumeration problem
initiated by mathematician Cayley Philos Mag 47(4):444–446, 1874] has been solved in such a systematic and integrated manner
as satisfying both mathematical and chemical requirements. |
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Keywords: | Planted 3D-tree Monosubstituted alkane Planted promolecule Cycle index Sphericity index Stereoisomer enumeration |
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