Symmetry properties of chemical graphs. VI. Isomerizations of octahedral complexes

Abstract

Isomerization of octahedral complexes XU,, by a mechanism known as diagonal twist (in which two ligands in cis positions, one to another, exchange sites) is considered. Construction of the corresponding isomerization graph is outlined (reported before by Balaban) and the problem of determining its symmetry is considered. Alternative routes for deducing the order of the automorphism group are described. The group is confirmed to be S , and representative symmetry permutations (one for each of the 11 classes of s6) are shown. Alternative pictorial representations of the 15-vertex graph are also shown.

On the Symmetry of a Zig-Zag and an Armchair Polyhex Carbon Nanotorus

“…We first solve a matrix equation by a MATLAB program [8] and then apply its output in a GAP program [9,10] Randic [11][12][13] showed that a molecular graph can be depicted in different ways such that its point group symmetry or three-dimensional (3D) perception may differ, but the underlying connectivity symmetry is still the same as characterized by the automorphism group of the graph.…”

Combinatorics of Petersen graph and its compositions for all irreducible representations for Jahn–Teller, non-rigid molecules and clusters

“…Thus there are two A 1 colorings for the term a 3 b 3 and 4 A 1 colorings for the a 2 b 2 cd term to color the pentagons of the hemi-dodecahedron with 4 colors. The corresponding numbers are (2,12), (2,16) and (0,8) for the G, H and T 1 = T 2 IRs of the A 5 group for the terms (a 3 b 3, , a 2 b 2 cd), respectively. We note that the two 3-dimensional IRs (T 1 and T 2 ) become degenerate for all GFs for coloring the pentagons in Fig.…”

“…Colorings of the nonplanar Petersen graph have also received attention, as it is a classic example serving as the smallest snark for the four-color problem, as the four-color theorem is synonymous with the result that no snark is planar. In chemical sciences, the Petersen graph represents the fluxional dynamics of isomerization reaction graph for the TBP complexes of transition metals and those of phosphorous and Group (15) elements, where the ligands undergo the Berry pseudorotation, as demonstrated by the pioneering works of Balaban [13][14][15][16], Mislow [17], King [21] and Randić [11,12]. Moreover when a degenerate electronic state couples to a vibrational mode of two-dimensional representation in the D 3h group, it undergoes the E⊗e Jahn-Teller coupling, and this in turn gives rise to dynamical distortions of TBP and octahedral complexes in doubly or triply-degenerate electronic states.…”

“…1. The Petersen graph can be drawn on the surface of a torus without any edges crossing, and thus it is toroidal, although its relative Desargues graph which also finds applications in isomerization reactions [11][12][13][14][15][16] is non-toroidal.…”

Symmetry properties of tetraammine platinum(II) with C2v and C4v point groups

“…The perception of the symmetry of a graph through the automorphism group of the graph had been studied in considerable depth (Balasubramanian, 1980;1981;1985;1995;2004a;2004b;2004c;Randic and Davis, 1984;Ezra, 1982;Herndon, 1983;Trinajstic, 1992), but the connection between the graph automorphism problem and the symmetry of a molecule has not been explored as much. Longuet-Higgins (1963) showed that a more desirable representation of molecular symmetry is to use the nuclear permutation and inversion operations resulting in a group called Permutation-Inversion (PI) group.…”