Graph-Theoretic Definitions

In this section we will define some of the important terms associated with the graphs in cluster expansion theory. 

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Using benzene-like aromatic systems and pericyclic reactions with an even number of centers, the principles of graph-theoretical structure theory are described and extended to conjugated heterocycles and cyclic systems with an odd number of centres. With topological analysis of the graphs of these systems as a foundation, a graph-theoretical definition of the idea of aromaticity in regard to monocyclic compounds is presented. 

The graph-theoretic definitions above are the only ones that are essential for the applications we will discuss below. Additional definitions may be found in more general discussions of graph theory. ... 

Although the graph-theoretical discussion of the aromatic sextet is not complete without formulating explicitly the definition of the super-sextet, we will skip over to the discussion of more chemical relevance [25],... 

Fig. 16. Example of tree-like representation for RNA secondary structure. Each hairpin structure is shown next to its equivalent tree. With such representations, a graph theoretic measure can measure the distance between these trees and help generate fitness values for a fitness landscape. For example, the distance between two structures may be defined as the minimal number of elementary graph operations (insert a point, switch an edge, etc.) needed to convert one tree into the other. Note that there are many variants of tree representations for RNA secondary structures and many definitions of graph distance. In low-resolution tree representations, several secondary structures can map to the same graph.

A Graph-Theoretic Study of the Numerical Solution of Sparse Positive-Definite Systems of Linear Equations. 

By generalization of this last definition of the Hosoya Z index to any graph and any graph-theoretical matrix M, the Hosoya-type indices were proposed as the sum of the absolute values of the coefficients of the characteristic polynomial of the matrix M. 

The objective of the molecular connectivity method is the quantitation of molecular structure based on the topological and electronic character of the atoms in the molecule. The pursuit of this objective leads to topological indexes derived from graph theoretic concepts, definitions, and procedures. We need to have methods for broad application in structure-property models, especially QSAR models. From these methods, we seek a basis for better understanding of the relation between structure and properties. 

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. 

Other graph-theoretical terms and definitions will be introduced below, and may also be found in books and reviews [23]. 

There are phenomena such as mesomerism or resonance that cannot be described in terms of our graph theoretical model of a molecule. The reason is that it is no longer possible to associate a unique and integral multiplicity to a covalent bond in aromatic structures. Instead, we introduce several resonance structures in order to represent aromatic compounds. Such structures are not generally isomorphic in the sense of Definition 1.17. 

Definition 2.5 A corohelicene (system) is a multiply connected, geometrically nonplanar (helicenic) polyhex, which is graph-theoretically planar. 

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. 

Graph Theoretic Approach. The multiscale segmentation behavior just described may efficiently be posed as a graph problem. As such, we will in this section review some basic graph theory and region definitions. 

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