Ambiguous molecular graphs

A node of an ambiguous molecular graph is colored by an element symbol or by an atom state, or by symbols that represent various alternatives. Typical symbols are A for any atom and Q for a hetero atom. 

Substructure restriction distance For two atoms i, 7 it an interval [a, b] c N of positive natural numbers is given that restricts the distance of the two atoms in the embedding of the ambiguous molecular graph into M. 

Substructure restriction hybridization For a nonempty subset [ij j e h Q k of atoms in an ambiguous molecular graph one introduces a hybridization tj for the atoms embedded via (p into the ambiguous molecular graph M. 

It is to be noted that the QSPR/QSAR analysis of nanosubstances based on elucidation of molecular structure by the molecular graph is ambiguous due to a large number of atoms involved in these molecular systems. Under such circumstances the chiral vector can be used as elucidation of structure of the carbon nanotubes (Toropov et al., 2007c). The SMILES-like representation information for nanomaterials is also able to provide reasonable good predictive models (Toropov and Leszczynski, 2006a). 

QTAIM provides a definitive answer to the question of whether two atoms are bonded or not even in ambiguous cases [87, 88] and, as a consequence, the molecular graph, i.e., the chemical structure, is readily defined by this theory. Two atoms are bonded if their nuclei are linked in space by a line of maximal electron density termed the bond path [89, 90] (see Fig. 1). A single bond path links the nuclei of chemically bonded atoms irrespective of the mode of the bonding covalent (single or multiple), hydrogen, van der Waals, ionic, metallic, etc. The properties of the eleetron density determined at the point of lowest density along the bond path, where... 

As previously mentioned, the acronym QSAR stands for the quantitative structure-activity relationship. However, there may be some ambiguity associated with the attribute quantitative. It does not necessarily follows that results expressed or having numerical representation are necessarily quantitative. Qualitative results can equally be numerically represented. Strictly speaking, we define and view QSAR models as quantitative only when the numerically expressed models allow meaningful interpretation of the numerical results obtained for the structure-activity relationship within the basic concepts of the particular model. This means that the physicochemical models should allow quantitative interpretation of the numerical physicochemical descriptors used and that the structure-mathematical models should allow quantitative interpretation of the numerical structure-mathematical descriptors used. We will use the symbol qsar and QSAR as the abbreviation for qualitative structure-activity relationship. Such are the relationships that are non-numerical and the relationships that may be numerical but the variables used are interrelated and thus do not allow unique interpretation of the MRA equations. Because all molecular descriptors hitherto used in QSAR, whether they are based on physicochemical properties, quantum mechanical calculations, or molecular graphs, are all interrelated, it follows that all such hitherto reported results, without further elaboration, remain essentially qualitative, being qsar rather than QSAR. 

The concept of infinite graph applicable to non-covalent molecular materials being less directly fruitful because of the ambiguity in the definition of the eigenvalue spectmm. 

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