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Symmetry properties of graphs of interest in chemistry. II. Desargues–Levi graph
Author(s) -
Randić Milan
Publication year - 1979
Publication title -
international journal of quantum chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.484
H-Index - 105
eISSN - 1097-461X
pISSN - 0020-7608
DOI - 10.1002/qua.560150610
Subject(s) - combinatorics , wreath product , mathematics , symmetry operation , chemistry , symmetric group , symmetry group , isomerization , graph , symmetry (geometry) , product (mathematics) , geometry , catalysis , biochemistry
The Desargues–Levi graph represents important chemical transformations: (1) isomerization routes for some carbonium ion rearrangements, (2) isomerization of trigonal bipyramidal structures, and (3) some pseudorotations of octahedral complexes. The symmetry properties of this graph have not been fully investigated in the past. Using the concept of the smallest binary code, all permutations which form the symmetry operations in the graph are registered. The resulting symmetry group can be represented as the direct product of S 5 (the full symmetric permutation group on five objects) and C i (the inversion in the center). There are 14 classes belonging to the following partitionings: 1 20 (1), 1 8 2 6 (1), 1 4 2 8 (1), 1 2 3 6 (1), 1 2 3 2 6 2 (1), 2 6 3 (2), 2 2 4 4 (2), 2 10 (3), 5 4 (1), and 10 2 (1). The total of 240 symmetry operations are distributed among the above 14 classes as follows: 1, 10, 15, 20, 20, 20, 20, 30, 30, 15, 10, 1, 24, and 24, respectively. Since partitioning cannot uniquely characterize a class, it is suggested that the distance between vertices in a cycle be introduced as an additional parameter to discriminate among classes having identical partitioning. Also, a suggestion to a generalization of the Mulliken notation for irreducible representations of the point molecular groups valid for more versatile symmetry groups of graphs is indicated.