Skeletal Atoms, the Topological State

Based on the use of chemical graph theory as described above, various indexes of molecular structure have been developed. These indexes may all be termed topological indexes. In the molecular connectivity method, indexes have been developed to characterize various aspects of molecular structure. The kappa shape indexes were developed so that shape measures could be directly entered in QSAR analyses. Each of these indexes characterizes the whole molecule with respect to one or more aspects of structure. In chemistry it is also of interest to characterize the skeletal atoms. In this final section we review briefly an investigation of the skeletal atoms as vertexes in the molecular graph as a basis for an atom descriptor. 

As a further development, Hall and Kier introduced the idea of the topological state of the graph vertex (which represents the skeletal atom). The topological state of the skeletal atom is the basis of both the determination of topological equivalence as well as computation of graphical indexes which can be related to the properties of the atoms in the molecule, The atom is viewed in the context of the full topology of the whole molecule. 

In the spirit of chemical graph theory, a numerical expression was developed to encode the structure information represented in the paths. The numerical value for the path from vertex i to vertex j is the entry f,y in a topological state matrix T. The topological state value, T, for vertex i is, then, the sum of the entries for row i. 

The key to useful topological state values is an appropriate form for the r, values. Hall and Kier have shown that simple forms, such as the graph distance d,j, are not useful because they fail to indicate proper topological equivalence. To ensure representation of topological equivalence, two features of the paths must be encoded (1) atomic identity and (2) the sequence of atoms in each path. It has been shown that both these characteristics can be encoded as follows. Atomic identity can be encoded using the molecular connectivity valence delta value, 8. The discussions concerning chi indexes and related quantities have shown the validity of the valence delta value as a characterization of atoms. 

It is not as easy to preserve the atom sequence in the paths. However, Hall and Kier have shown that topological equivalence can be preserved when the 

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