Geometry matrix

Autocorrelation descriptors calculated for 3D-spatial molecular geometry are based on interatomic distances collected in the -> geometry matrix and the property function is defined by the set of atomic properties. 

The 3D-Balaban index was derived from the -> geometry matrix G as ... 

Similar to the previously defined connectivity indices but relative to the -> geometry matrix G, they are defined using the -> geometric distance degree in place of the topological vertex degree 6 ... 

The determinant of (A" -i- G) was also proposed as a molecular topographic descriptor where A is the weighted adjacency matrix (- weighted matrices) where the entries corresponding to bonded atoms are - bond distances and G the geometry matrix [Mihalic et ai, 1992a]. 

A square symmetric matrix Ax A, A being the number of molecule atoms, whose entries are the quotient of the corresponding elements of the molecular geometry matrix G and the graph -> distance matrix D therefore, each entry is defined as ... 

The basis of the DG method is that each ligand molecule is represented as a collection of points in space, each corresponding to atoms or groups of atoms. For a chosen molecular representation, the conformation of the molecule is described in terms of distance between points, i.e. -> interatomic distances. The matrix containing interatomic distances between all possible pairs of points is the -> geometry matrix of the molecule. To account for molecular flexibility, a matrix of lower bounds on the considered interatomic distances and a matrix of upper bounds are also defined fixed interatomic distances are represented by equal values in these matrices. It should be noted that the calculation of the atomic coordinates is a really difficult problem, and several mathematical techniques have been proposed [Crippen, 1977 Crippen, 1978 Crippen, 1991],... 

In general, geometrical descriptors can be derived from the molecular matrix, Z-matrix or geometry matrix. 

The geometry matrix G (or geometric distance matrix) of a molecule, obtained from the molecular matrix M, is a square symmetric matrix Ax 4 whose entry / is the geometric distance calculated as the Euclidean distance between the atoms s and t. 

The reciprocal geometry matrix is obtained from the geometry matrix as the following ... 

From the geometry matrix, the usual -> graph invariants can be calculated such as -t characteristic polynomial, -> eigenvalue-based descriptors, -> path counts, - ID numbers, -> 3D-Balaban index, -> 3D-Schultz index and so forth [Randic, 1988b Nikolic et al, 1991]. It is noteworthy that all these indices are sensitive to molecular geometry. Moreover, the geometry matrix is used for the calculation of size descriptors and - 3D-MoRSE descriptors. 

Important derived matrices are the powers of the geometry matrix, used to define - molecular profiles descriptors, and the -> distance/distance matrix DD, which unifies the topological and geometric molecular information. 

Molecular descriptors derived from the -> distance distribution moments of the - geometry matrix G, defined as the average row sum of its entries raised to the kXh power, normalized by the factor k ... 

In this case, the row siuns of the geometry matrix are obtained by summing only the geometric distance powers of the atoms belonging to the periphery, and the average is found from the number of the contributing atoms only. Each atomic distance sum is considered as a local indicator of molecular shape, and each molecular invariant S is considered a global shape descriptor. 

Bond profiles constitute a generalization of atomic molecular profiles since they provide a characterization of molecular connectivity, which is not explicitly contained in the geometry matrix [Randic, 1996a Randic and Krilov, 1996]. 

The geometrical shape coefficient /a is calculated in the same way but using the information of the -> geometry matrix G [Bath et al, 1995] ... 

A Wiener-type index, called 3D-Wiener index, is derived from the - geometry matrix G as a A... 

The Euclidean-distance map matrix, denoted as ED, is the analogue of the geometry matrix G derived from a molecular graph. In this case, vertices of the map graph are assigned x, y) or x, y, z) coordinates, z being intended as the weighting scheme for vertices it is defined as [Bajzer, Randic et al, 2003]... 

Table C2 Some Hosoya-type indices for the data set of 18 octane isomers (Appendix C - Set 1) calculated on the unweighted molecular graph and the following graph-theoretical matrices A, adjacency matrix D, distance matrix L, Lapiacian matrix D , reciprocal distance matrix % matrix C reciprocal geometry matrix.

Characteristic polynomial, spectrum, spectral moments, eigenvectors, and Hosoya-type indices were also computed on square molecular matrices encoding information about spatial interatomic distances such as the geometry matrix G and the reciprocal geometry matrix [Ivanciuc and Balaban, 1999c]. 

Moreover, following the same procedure, other expanded matrices are defined [Minailiuc, Katona et al., 1998] replacing the topological distance matrix D by the —> geometry matrix G as... 

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