Topological distance matrix

The derivation of the topological distance matrix from the molecular graph is followed by the assignment of PPPs to the nodes of the graph. The following list provides chemical definitions of the five PPP types that are implemented in the CATS descriptor. The upper-case letter in parentheses is the abbreviation of each PPP type. Additionally, a functional group description is paired with its corresponding SMARTS in square brackets ... 

A numerical value associated with chemical constitution that can be used to correlate chemical structure with various physical properties, chemical reactivity, or biological reactivity. The numerical basis tor topological indices is provided (depending on how a molecular graph is converted into a numerical value) by either the adjacency matrix or the topological distance matrix. In the latter, the topological distance between two vertices is the number of edges in the shortest path between these. 

In the topological distance matrix, the topological distance between two vertices is the number of edges in the shortest path between these. 

These are deduced from a topological picture (2D picture) of the molecules. The picture carries information on how the atoms are connected and what is the nature of bonds (structural formula of a molecule). Mathematically, the topology picture is described with the connectivity matrix. Pioneering work in this field was published in 1947 by Wiener on paraffin hydrocarbons [31]. It is defined as a half sum of the off-diagonal elements in the topological distance matrix. In the last few decades dozens of descriptors have been deduced... 

Some molecular descriptors, called - determinant-based descriptors, are calculated as the determinant of a - matrix representation of a molecular structure. Moreover, permanents, short- and long-hafnians, calculated on the topological - distance matrix D, were used as graph invariants by Schultz and called per(D) index, shaf(D) index, lhaf(D) index [Schultz et al, 1992 Schultz and Schultz, 1992]. 

It is a square unsymmetric Ax A matrix, where the upper triangle of the matrix contains the elements of the detour matrix (information about the longest paths) and the lower triangle contains the elements of the topological distance matrix (information about the shortest paths). 

The largest eigenvalue of the topological - distance matrix D representing an H-depleted molecular graph was proposed as a molecular descriptor, and was called the leading eigenvalue of the distance matrix [Schultz et al., 1990]. It was found to be a... 

It is the largest eigenvalue of the matrix obtained as the sum of the adjacency matrix A and the topological distance matrix D representing a molecular graph [Schultz et al., 1990]. Its logarithm was used to model physico-chemical properties [Cash, 1995c]. 

For characterization of 2D structures, molecular profiles are computed in the same way by the distance distribution moments of the topological distance matrix D. 

Rouvray, D.E. (1986). The Role of the Topological Distance Matrix in Chemistry. In Mathematical and Computational Concepts in Chemistry (Trinajstic, N., ed.), Ellis Horwood, Chichester (UK), pp. 295-306. 

Calculation of the Balaban distance connectivity index] and Jt index for 2-methylpentane. D is the topological distance matrix distance sums and the vertex degrees. B equals 5 and C is zero. 

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... 

Selecting different combinations of Mi and M2 matrices leads to the derivation of several Schultz-type indices. The original Schultz molecular topological index MTI is obtained for Ml = A and M2 = D, where D is the topological —> distance matrix. Typical Schultz indices are derived from (D, A, D), A, D ), (W, A, D), (W A, D ), (W, A, W), (UCJ, A, UCJ), (USZ, A, USZ), where is the reciprocal distance matrix, W is one among —> walk matrices, is the reciprocal walk matrix, UCJ and USZ the unsymmetrical Cluj and Szeged matrices, respectively. 

Values of some indices derived from Wiener matrices Wg and Wp for the H-depleted molecular graph of 2-methylpentane D is the topological distance matrix. 

We consider here two kinds of distance/distance matrices the topographic distance/topological distance matrix and the corresponding reciprocal matrix. We denote the first matrix as D/ D and the second as D/ D. 

The topographic distance/topological distance matrix is an unsymmetric VxV matrix, in which the upper matrix-triangle represents the same part of the corresponding topographic matrix, and the lower matrix-triangle the same part of vertex-distance matrix (defined in Section 4.1) ... 

The topographic distance/topological distance matrix D/D of T2 is given by... 

A broad category of topological indices are calculated based on the values of the Detour Maximal Topological Distance) Matrix (Ivanciuc and Balaban 1994). Recently, new descriptors have been proposed in this category, which are useful for evaluating solubility in water (Talevi et al. 2006). 

Those based on the topological distance matrix, including the Wiener index, the polarity number, the distance sum, the Altenburg polynomial, the mean square distance, the Hosoya index, and the distance polynomial ... 

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