Bond-path distance matrix

FIGURE 4.1 Matrices can be used for describing chemical structures in different fashions. The adjacency matrix (a) of thionyl chloride shows whether are bonded to each other. The distance matrix (b) describes the number of bonds between two elements of a structure. The Cartesian distance matrix (c) contains the real three-dimensional (Euclidean) distances between atoms calculated from Cartesian coordinates of the atom positions. The bond path distance matrix (d) contains the sum of bond length between two atoms and is, in contrast to the Cartesian matrix, independent of the conformation of the molecule. 

We have seen before that different types of matrices can be used for characterizing a molecule. Depending on which matrix is used, the distance r j in a radial function can represent either the Cartesian distance, a bond-path distance, or simply the number of bonds between two atoms. Consequently, we yield three groups of RDF descriptors. 

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. 

The Cluj-distance index CJD is obtained from the symmetric Cluj-distance matrix CJD as the sum of the matrix entries corresponding to all the P order paths (i.e. bonds) above the main diagonal, i.e. applying the Wiener operator "iPto the P order sparse symmetric Cluj-distance matrix CJD or from the sparse unsymmetric Cluj-distance matrix CJDjj applying the Wiener orthogonal operator "W ... 

Distance Matrix is a chemical structure representation in a matrix consisting of either Euclidean distances or the sum distances along the shortest bond path between two atoms. 

The path-distance map matrix, denoted as PD, resembling the bond length-weighted distance matrix of a molecular graph, is defined as [Bajzer, Randic et al, 2003]... 

The chemical distance matrix is a variant of the multigraph distance matrix defined by using the —> chemical distance as the weighting scheme for edges therefore, the path weight Wy in terms of bond chemical distances is calculated as [Balaban, Bonchev et al., 1993] ... 

The first topological index to be published is the Wiener index W, which is half the sum of the bond-by-bond path lengths between each atom [41]. It is easily calculated from each offdiagonal element Dtj of the distance matrix of a structure ... 

The first index based on a graph approach to molecular structure was developed by Wiener in 1947. The path number was defined as the number of bonds between all pairs of atoms in an acyclic molecule. Using the path number and another index, Wiener was able to fit alkane boiling points fairly well. The Wiener number is inversely proportional to the compactness of a molecule. In 1971 Hosoya connected the Wiener number with graph theory, pointing out that the Wiener number is the half-sum of all the distance matrix entries for a molecule. ... 

If one assumes that all bonds have the same (unit) resistance one can write a resistance-distance matrix 0. This matrix has also been referred to as a Kirchhoff matrix, in view of the fact that it rests on Kirchhoff s current flow laws. The resistance-distance matrix better reflects interatomic distances in cyclic compounds than the ordinary distance matrix, as it takes into account not only the shortest paths in a graph but also the presence of alternative connections between vertices. 

Half-sum of the cube of the adjacency matrix Polarity number number of third neighbors Gordon-Scantlebury index number of second neighbors Number of paths of length /i = 0-10 Balaban s J index based on distance Balaban s J index based on multigraph bond orders Balaban s J index based on relative electronegativities Balaban s 7 index based on relative covalent radii Balaban s information-based indexes on distance sums... 

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