Point symmetry graph

Based on the above neighbor relation, and by direct analogy with the model of ref.[20], we define the point symmetry graph g(M,sym) of the nuclear configuration space M of the given stoichiometric family of chemical species by the following relations ... 

The vertex set V(g(M,sym)) and edge set E(g(M,sym)) define the graph g(M,sym). This graph provides a concise representation of the mutual arrangements of point symmetry domains Gy within the nuclear configuration space M. 

Such a procedure corresponding to the permutation-nomenclature system of configuration 7> is in accordance with the fact that a chemical constitution is a representative of a product D, X. .. X Da of the double cosets D, of pairs of subgroups symmetric groups Sn,g. The latter correspond to permutations of atoms of the same element and the symmetry of the adjacent subset of graph points whose degree is compatible with the coordination number of the element equivalence class of the atom. 

As if these synthetic symmetries were not enough, in addition we find that when the first order of difference of the King Wen sequence is graphed it appears random or unpredictable (fig. 18A). Elowever, when an image of the graph is rotated 180 degrees within the plane and superimposed upon itself, it is found to achieve closure at four adjacent points (fig. 18B). 

Rtlcker, G. and Rucker, C. (1991a). Isocodal and Isospectral Points, Edges, and Pairs in Graphs and How to Cope with Them in Computerized Symmetry Recognition. J.Chem.Inf.Com-putScL,31,422-421. 

The alternative description of the 4 5 and 5 7 uranyl molybdate frameworks can be made in terms of tubes or tubular units fused into a 3D framework. Such tubular units for the 4 5 and 5 7 structures are shown in Fig. 60. The topological structure of these tubes can be exposed by unfolding them onto a 2D plane so that the result is a ID graph with local topology identical to that of the tube. To reform the tubular unit, one has to fold the tape and to join the points identified by the same letters. The tubular graphs shown in Figs. 60b and d symbolize channels in the uranyl molybdate frameworks. It is noteworthy that they have a chiral helical structure that is in agreement with the symmetry of the chaimels in the real structures (6522 for 4 5 and 65 for 5 7 frameworks). 

Rucker, G. and Rucker, C. (1991a) Isocodal and isospectral points, edges, and pairs in graphs and how to cope with them in computerized symmetry recognition./. Chem. Inf. Comput. Sci., 31,422 27. 

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