Graph constitutional symmetry

An encoding of a molecular structure based on its connection table, which represents only the constitution of a molecule without consideration of atom coordinates, bond lengths, or bond angles. See Canonical Numbering and Constitutional Symmetry Graph Theory in Chemistry and Structure Representation. 

The Morgan algorithm 5> is a device used by the Chemical Abstracts Service for assigning indices to the nodes of constitutional formulas whose H-atoms have been omitted, i. e. their reduced graphs. If the indices of all atoms are needed, e.g. for the representation of stereochemistry, an additional procedure is needed for establishing those for the H-atoms. Constitutional symmetry is not indicated directly by Morgan indices. 

See, for example, M. Razinger, K. Balasubramanian, and M. E. Munk,/. Chem. Inf. Comput. Sci., 33, 197 (1993). Graph Automorphism Perception Algorithms in Computer-Enhanced Structure Elucidation. C. Jochum and J. Gasteiger, J. Chem. Inf. Comput. Sci., 17, 113 (1977). Canonical Numbering and Constitutional Symmetry. See also Ref. 7... 

The problem of canonical coding, graph isomorphism, and graph automorphism has both mathematical and chemical significance. The mathematical formulation of the problem is briefly set out below, and some cormections with the chemical counterpart are presented. In the subsequent sections, the main algorithms used in chemistry for canonical coding of molecular graphs and constitutional symmetry perception are presented and compared. 

From the above properties of the permutations E, A, B, and C, one may conclude that they form the automorphism group of the graph (5). This automorphism group describes all constitutional symmetry relationships of isopropylcyclopropane. 

This number can be further reduced by the use of more powerful vertex invariants, like the distance sum. The distance sum, DS, of a vertex i is the sum of the topological distances from vertex i to all other vertices in the molecular graph. The partitioning in the DS equivalence classes of atoms from 4-ethyloctane is presented in (10), showing that only two atoms have identical DS values. Using this partitioning, the constitutional symmetry of 4-ethyloctane can be determined with 1 1 1 1 1 1 1 1 2 = 2 permutation labelings. 

Atoms characterized by identical NOON values are considered to belong to the same equivalence class. As is apparent from the NOONs computed for graph (22) some nonequivalent atoms receive identical values. This initial partition is refined using a set of rules and an iterative algorithm. The heuristic partitioning of atoms in classes is used in connection with an automorphism detection CCAP algorithm, which makes the whole procedure efficient for canonical coding and perception of constitutional symmetry. 

Canonical Numbering and Constitutional Symmetry Combinatorial Chemistry De Novo Ligand Design Diversity of Chemical Libraries Graph Theory in Chemistry Symmetry in Chemistry. 

See Canonical Numbering and Constitutional Symmetry and Graph Theory in Chemistry. 

Canonical Numbering and Constitutional Symmetry Chemical Abstracts Service Information System Graph Theory in Chemistry Inorganic Three-dimensional Structure Databases Polymers Structural Representation Standard Exchange Formats for Spectral Data Structure and Substructure Searching Structure Databases Structure Representation Three-dimensional Structure Generation Automation Three-dimensional Structure Searching. 

It is further possible to describe a given constitution by mapping the indexed set of atoms onto the independently indexed graph. The indexed set of atoms must be partially symmetrized with regard to the element equivalence classes of the atoms and the indices of the graph with respect to its symmetry. That constitution is defined as the reference constitution in which the atomic indices a and the graph indices g match. 

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. 

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