Graphs, molecular

Central the molecular graph is completely coded (each atom and bond is represented) matrix algebra can be used the niimber of entries in the matrix grows with the square of the number of atoms in ) no stereochemistry included... 

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. 

Accordingly, a molecular structure can be represented by the molecular graph of Eq. (2). 

The Wiener index was originally defined only for acyclic graphs and was initially called the path number [6]. "The path number, W, is defined as the sum of the distances between any two carbon atoms in the molecule in terms of carbon-carbon bonds". Hosoya extended the Wiener index and defined it as the half-sum of the off diagonal elements of a distance matrix D in the hydrogen-depleted molecular graph of Eq, (15), where dij is an element of the distance matrix D and gives the shortest path between atoms i and j. 

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. 

Covalen t radii for all th e clem cn ts are readily available an d the bond orders of all bonds arc available from the molecular graph. Prior to describing the explicit default parameter scheme, it is nee-... 

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. 

Many of the descriptors which can be calculated from the 2D structure rely upon the molecular graph representation because of the need for rapid calculations. Kier and Hall have developed a large number of topological indices, each of which characterises the molecular structure as a single number [Hall and Kier 1991]. Every non-hydrogen atom ir the molecule is characterised by two delta values, the simple delta Si and the valence delta SJ ... 

The number of neighbors is given by the molecular graph and the following rules determine a hybridization state for each atom in a molecule. 

MOLMOL (MOLecule analysis and MOLecule display) A molecular graph-... 

Fig. 1.32. (a) Molecular graphs and electron density contours for pentane and hexane. Dots on bond paths represent critical points, (b) Comparison of molecular graphs for bicycloalkanes and corresponding propellanes. (Reproduced from Chem. Rev. 91 893 (1991) with permission of the American Chemical Society.)... 

An alternative stream came from the valence bond (VB) theory. Ovchinnikov judged the ground-state spin for the alternant diradicals by half the difference between the number of starred and unstarred ir-sites, i.e., S = (n -n)l2 [72]. It is the simplest way to predict the spin preference of ground states just on the basis of the molecular graph theory, and in many cases its results are parallel to those obtained from the NBMO analysis and from the sophisticated MO or DFT (density functional theory) calculations. However, this simple VB rule cannot be applied to the non-alternate diradicals. The exact solutions of semi-empirical VB, Hubbard, and PPP models shed light on the nature of spin correlation [37, 73-77]. 

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 adjacency matrix A(G) of a molecular graph G with N vertices is the square NxN symmetric matrix in which [A],j=l if vertex Vj is adjacent to vertex Vj and [A],j=0 otherwise. The adjacency matrix is symmetric, with all elements on the main diagonal equal to zero. The sum of entries over row i or column i in A(G) is the degree of vertex Vj, 5j. Usually, the adjacency matrix is based on weighted molecular graphs in which heteroatoms are represented as vertex parameters and... 

The second contribution to the E-state index comes from the interactions between an atom i and all other atoms in the molecular graph. The perturbation on the intrinsic state value / of atom i due to another atomj depends on the difference between the corresponding intrinsic state values, (f- /,), and on the graph distance between atoms i andj. The overall perturbation on the intrinsic state value I of atom i is ... 

Ivanduc, O., Ivanduc, T. Matrices and structural descriptors computed from molecular graph distances. In Topological Indices and Related Descriptors in QSAR and QSPR, Devillers, J. Balahan, A. T. (eds.), Gordon Breach, Amsterdam, 1999, pp. 221-277. 

D., Balahan, A. T. Comparison of weighting schemes for molecular graph descriptors application in quantitative structure-retention relationship models for alkylphenols in gas-liquid chromatography. J. Chem. Inf. Comput. Sci. 2000, 40, ITl-lM,. 

Hall, L. H., Mohney, B Kier, L B. The electrotopological state Structure information at the atomic level for molecular graphs. J. Chem. Inf. Comput. Set. 1991, 31, 76-82. 

Ovidiu Ivanciuc describes the computation of Electrotopological State (E-state) Indices from the molecular graph and their application in drug design. The E-state encodes at the atomic level information regarding electronic state and topo-... 

A.T. Balaban, ed.. Chemical Applications of Graph Theory, Academic Press, London, 1976. V. Kvasnicka and J. Pospichal, An improved version of the constructive enumeration of molecular graphs with described sequence of valence states. Chemom. Intell. Lab. Systems, 18 (1993) 171-181. 

---