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Symmetry properties of chemical graphs. V. Internal rotation in XY ⋅ 3 XY ⋅ 2 XY 3
Author(s) -
Randić Milan
Publication year - 1983
Publication title -
journal of computational chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.907
H-Index - 188
eISSN - 1096-987X
pISSN - 0192-8651
DOI - 10.1002/jcc.540040112
Subject(s) - combinatorics , wreath product , mathematics , isomorphism (crystallography) , graph isomorphism , graph , symmetry group , order (exchange) , group (periodic table) , semidirect product , symmetric group , discrete mathematics , product (mathematics) , chemistry , crystallography , physics , line graph , quantum mechanics , geometry , crystal structure , finance , economics
Structures XY ⋅ 3 XY ⋅ 2 XY 3 of symmetry C 2 v (of which propane is an example) are examined and the rearrangement due to the internal rotation of the end groups XY 3 studied. The isomerization graph is constructed, various forms of which are displayed and the symmetry of which has been determined. The order of the group is 72. There are nine prime (irreducible) representations (4A + E + 4G) with the following partitioning of the elements into classes: 1, 4 2 , 6 2 , 9, 12 2 , 18. When the mechanism for rearrangement is generalized to include enantiomers, a duplex graph is produced with the order of the group 144 which is isomorphic to the group S 2 ( S 3 , S 2 ) (generalized wreath product of the symmetric group S 2 and S 3 ). The corresponding graph has been constructed and displayed in one of more symmetrical forms. Isomorphism of groups of order 144 is discussed and a procedure is outlined in which correspondence between distinctive combinatorial objects is established by inducing permutations of m elements from available permutations of n elements. The scheme is based on selection of suitable graph invariants in one system and their labeling as m objects which form the basis for representation of the symmetry for the other system.