Connected molecular graph

For the existence of at least one connected molecular graph we also need (Con) Zxe Zxe P(X)+2 > 0. This condition is necessary, since a connected... 

The given arguments show that the three conditions are necessary for the existence of a connected molecular graph. The proof that they are also sufficient is more important and complicated, we refer to Grund [95]. We denote the set of molecular formulas satisfying (Grl), (Gr2) and (Con) by... 

For the existence of a connected molecular graph we need in addition that the following holds ... 

Moreover, we assume a transversal T of this set of labeled connected molecular graphs, together with a function... 

A major disadvantage of a matrix representation for a molecular graph is that the number of entries increases with the square of the number of atoms in the molecule. What is needed is a representation of a molecular graph where the number of entries increases only as a linear function of the number of atoms in the molecule. Such a representation can be obtained by listing, in tabular form only the atoms and the bonds of a molecular structure. In this case, the indices of the row and column of a matrix entry can be used for identifying an entry. In essence, one has to distinguish each atom and each bond in a molecule. This is achieved by a list of the atoms and a list of the bonds giving the coimections between the atoms. Such a representation is called a connection table (CT). 

A connection table has been the predominant form of chemical structure representation in computer systems since the early 1980s and it is an alternative way of representing a molecular graph. Graph theory methods can equally well be applied to connection table representations of a molecule. 

A molecule is represented by a tree which Rarey and Dixon called a feature tree, within which the nodes are fragments of the molecule. The atoms belonging to one node are connected in the molecular graph. A node consists at least of one atom. 

Edges in the feature tree connect two nodes which have atoms in common or which have atoms connected in the molecular graph. Rings are collapsed into single nodes. 

There are a number of different ways that the molecular graph can be conununicated between the computer and the end-user. One common representation is the connection table, of which there are various flavours, but most provide information about the atoms present in the molecule and their connectivity. The most basic connection tables simply indicate the atomic number of each atom and which atoms form each bond others may include information about the atom hybridisation state and the bond order. Hydrogens may be included or they may be imphed. In addition, information about the atomic coordinates (for the standard two-dimensional chemical drawing or for the three-dimensional conformation) can be included. The connection table for acetic acid in one of the most popular formats, the Molecular Design mol format [Dalby et al. 1992], is shown in Figure 12.3. 

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... 

Toropov AA, Toropova AP, Nesterova AI, Nabiev OM (2004a) Prediction of alkane enthalpies by means of correlation weighting of Morgan extended connectivity in molecular graphs. Chem. Phys. Lett. 384 357-363. 

Bond critical points represent extremes of electronic density. For this reason, these points are located in space where the gradient vector V p vanishes. Then the two gradient paths, each of which starts at the bond critical point and ends at a nucleus, will be the atomic interaction line. When all the forces on all the nuclei vanish, the atomic interaction line represents a bond path. In practice, this line connects two nuclei which can consequently be called bonded [5]. In terms of topological analysis of the electron density, these critical points and paths of maximum electron density (atomic interaction lines) yield a molecular graph, which is a good representation of the bonding interactions. 

The basic mathematical structure which maps a certain molecule (molecular graph G) is the adjacency matrix of the molecular graph A(G). For a molecule having N atoms, A(G) is a square N x N matrix. Its entries ay have only two different values 1, and 0, due to the fact that two atoms in a molecule are in binary relation, being either connected or not connected ... 

Delta Value Schemes and Molecular Connectivity Indices A delta (6 or 6V) value is an atomic descriptor for nonhydrogen atoms in a molecular graph. The superscript-free delta value, <5, is defined as the number of adjacent nonhydrogen atoms of atom i ... 

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