Skeletal molecular graph

According to Graph Theory, a molecule may be represented by its skeletal molecular graph which, from a mathematical point of view, is the union of a set of points, symbolising the atoms other than hydrogens, and a set of lines, symbolising bonds. Its properties can be then expressed in terms of graph-theoretical invariants, Njj, which have been defined as "the number of distinct ways in which skeleton i... 

The Bertz index incorporates skeletal complexity by representing the molecular structure as a graph (a skeletal molecular graph that represents all nonhydrogen atoms) that will have properties that are expressed as so-called graph theoretical invariants. Bertz uses information theory to incorporate the effects of symmetry, and also accounts for molecular size and the presence of heteroatoms in his general index of complexity. 

This development is appropriate only for saturated alkanes - - because there is no provision for atom identity, bond types, or number of hydrogen atoms in each skeletal group (except for the case of saturated hydrocarbons.) This branching index is essentially a mathematical property of graphs it does not adequately represent molecular graphs. Further, a single index would appear to be insufficient to relate to the wide variety of molecular properties, especially when biological and environmental properties are of interest. 

In the molecular connectivity method a series of indexes is developed, expressing structure information at several levels. The molecular skeleton is conceived as consisting of fragments of different sizes and complexity. The molecular graph may be decomposed into fragments called subgraphs, such as a skeletal bond, a pair of adjacent bonds, a cluster of bonds to a central atom, etc. 

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. 

The final step in the derivation of the electrotopological state is the combination of the atom intrinsic state expression with the experession for the perturbation by all other skeletal atoms. The perturbing influence on atom i by atom j was assumed to be proportional to the difference between the atom intrinsic values, U — Iy Further, the perturbation falls off as the square of the distance between the two atoms. The distance, in this case, is taken to be the number of atoms in the (shortest) path between atoms i and jj namely Vij. The overall perturbation on atom i, A/ is taken as a sum over all other atoms in the skeleton (molecular graph) ... 

The problems experienced with similarity queries with low chemical specificity identify a need for 2-D attribute sets which consider larger connectivity pattern units of the molecular graph. Indeed, they should describe the skeletal graph (i.e., a graph devoid of chemical embellishment) as a complete pattern unit. In 3-D, the molecular shape is completely defined by the full (A) distance matrix (e.g., Crippen ) involving all atom pairs in the structure, whether they are bonded or not. In 2-D, a similar description can be expressed (e.g., Bersohn ) by considering the minimum number of bonds that must be traversed to reach atom j from atom i. A number of authors (e.g., Randic and Wilkins Carhart, Smith and Ven-kataraghavan ) have used these inter-nodal bond separations (INBSs) to derive atom codes for structure-activity studies. They would also seem to form a valid basis for 2-D similarity analyses. 

The approach used in chemical graph theory is to abstract from the molecular structure those elements that lead to structure variables in the form of numerical indexes. The set of atoms and connections is viewed as structure information but in a form not amenable directly to QSAR analysis. The first step is to adopt a form for the molecular skeleton as the basis for extraction of structure information. To represent the molecular skeleton, the hydrogen-suppressed graph is most commonly used hydrogen atoms are not explicitly considered hydrogen atoms are incorporated in skeletal groups which are the graph vertexes. 

The identifying characteristics of atoms include atomic number and number of electrons partitioned between valence electrons and core electrons. The immediate bonding environment of atoms in the molecular skeleton depends on the number and arrangement of the valence electrons and the number and type of bonds. In most graph theoretical methods, a hydrogen-suppressed skeleton is used to facilitate the counting and enumeration of skeletal features. 

It is also possible to use chemical graph theory to characterize molecular shape and the individual skeletal atoms. However, before those descriptions are developed, several applications of molecular connectivity chi indexes will be presented. 

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