Graph theory definition

Table 2-4. Some basic definitions for graph theory. Graph theory term...

Sec. 4, and the definitions in graph theory discussed in the previous chapter (Sec. 36). 

The E-state is based solely on atom connectivity information obtained from the molecular graph, without any input from the molecular geometry or sophisticated quantum calculations. We start this chapter with a brief presentation of the relevant notions of graph theory and continue with the definitions of a couple of important graph matrices. Then the molecular connectivity indices are mentioned... 

The formalism of graph theory lends itself to a number of very useful definitions. One useful concept is the degree d(v) of a vertex v, which is defined as the number of edges with which the vertex is incident. Another is a... 

The analysis, however, shows that, even when all the factors in the denominator of eqn. (46) are Arrhenius factors, reaction rate constants cannot always be determined on their basis. The analysis carried out using graph theory methods shows that it is possible only for definite types of mechanisms, namely for those that correspond to (a) Hamiltonian or (b) strong bi-connected graphs (the latter term is due to Evstigneev) [54]. 

A gra[. is defined as a set of elements V, with a binary relation E defin on the set- Usually circles represent vertices ( elements) and lines, usually referred to as edges, represent binary relations- The binary relation is both symmetric and anti reflexive- The classical text of Harary s Graph Theory [ll] collects several of the more popular types of graphs- One observes the inherent generality of the definition of graphs There are no restrictions on what the set of elements, V, represents or to what the binary relation, E, corresponds- If we model a chemical structure, V, may be the set of atoms, while E the set of bonds present, but equally E may represent "close" atomic contacts (such as those occuring in crowd molecules, etc-)- Atlernatively V... 

Mathematics and statistics, graph theory, computational chemistry and molecular modelling techniques enable the definition of a large number of theoretical descriptors characterizing physico-chemical and biological properties, reactivity, shape, steric hindrance, etc. of the whole molecule, molecular fragments and substituents. 

Some basic concepts and definitions of statistics, chemometrics, algebra, graph theory, similarity/diversity, which are fundamental tools in the development and application of molecular descriptors, are also presented in the Handbook in some detail. More attention has been paid to information content, multivariate correlation, model complexity, variable selection, and parameters for model quality estimation, as these are the characteristic components of modern QSAR/QSPR modelling. 

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. 

Noting that faces are exactly what in graph theory are called cycles, the previous Euler formula provides the definition of —> cyclomatic number C ... 

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