Graph-Theoretical Reference Structure

Similarly, Figures 11 and 12 show that the Hess— Schaad and graph-theoretical reference structures give very similar predictions of aromaticity for the annulenes. Without a more precise experimental definition that is generally accepted, we cannot say which of these two reference structures is better. Both appear to give at least roughly correct predictions of aromaticity. 

Figure 12. Resonance energies per jt electron in units of P for the conjugated monocyclic hydrocarbons using the graph-theoretical reference structure and the Hiickel com putational method.

We will use a topological formalism to develop most of our structure-property correlations for polymers. This formalism utilizes connectivity indices defined via graph theoretical concepts as its main structural and topological descriptors. Connectivity indices have been widely used for simple molecules. A review is provided in this section to familiarize the reader with these indices before discussing their extension to polymers. The information in this section is summarized from two books by Kier and Hall [1,2], to which the reader is referred for additional details. The first book [1] is more detailed, while the second book [2] includes the results of the research over the decade after the publication of the first book. 

Graph-theoretical algorithms and data structures provide the basis for all modem 2-D chemical information systems, which offer three main types of searching facility. Structure search involves the search of a file of compounds for the presence or absence of a specified query compound. Such a search is required when there is a need to retrieve data associated with some compound or when a new molecule is to be added to a database and one needs to establish that it is not already present (a process that is normally referred to as registration). Substructure search involves the search of a file of compounds for all molecules containing some specified query substructure, irrespective of the environment in which the query substructure occurs. Finally, similarity search involves the search of a file of compounds for those molecules that are most similar to an input query molecule, using some quantitative definition of structural similarity. These three types of retrieval mechanism are considered now. 

The permanent (also referred to as the positive determinant) of the vertex-adjacency matrix per A can be used to enumerate the number of Kekule structures K, or in the graph-theoretical terminology 1-factors (Harary, 1971 Cvetkovid et al., 1995) or dimers (Percus, 1969, 1971 Cvetkovid et al., 1995), of alternant structures (Mine, 1978 Cvetkovid et al., 1972, 1974a Kasum et al., 1981 Schultz et al., 1992 Cash, 1995 Torrens, 2002 Jiang et al., 2006) ... 

Most of the graph-theoretical matrices presented have been used as sources of molecular descriptors usually referred to as topological indices. They are particularly concerned with a special class of graphs that represents chemical structures involving molecules. Due to its multidisciplinary scope, this book will appeal to a broad audience ranging from chemistry and mathematics to pharmacology. 

In addition to structural measures of position within whole and ego networks, there are also a number of measures that describe the overall structural properties of a network (whether a whole network or an ego network). Figure 44.6 shows two illustrative networks and reports three commonly used structural measures for each density (which assesses the total number of ties relative to the total number of possible ties) size (which is a count of the total number of ties in a network) and network centralization (which assesses the extent to which the ties within a network are shared across nodes versus centralized in a few nodes). For precise mathematical formulas and more detailed explanations of the logic behind these and the many other graph-theoretic measures—such as core-peripheriness, dumpiness, scale-freeness— we refer the interested reader to Wasserman and Faust (1994) and Carrington, Scott, and Wasserman (2005). 

Mathematical representation of a structure allows one to extract from a structure a number of mathematical invariants, quantities that do not depend on assumed labeling of atoms in the structure. One refers to structural invariants as topological or topographic indices if they reflect upon the molecular connectivity or molecular geometry, respectively. The former are based on a graph as a model for a molecule. Hence, correctly these invariants should have been called graph theoretical indices. However, the label topological index prevailed. 

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