Symmetry properties of graphs of interest in chemistry. II. Desargues–Levi graph

Abstract

AbstractThe 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 S5 (the full symmetric permutation group on five objects) and Ci (the inversion in the center). There are 14 classes belonging to the following partitionings: 120(1), 1826(1), 1428(1), 1236(1), 123262(1), 2 63(2), 2244(2), 210(3), 54(1), and 102(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.