The Distance-Sum-Connectivity Matrix

One can generate the distance-sum-connectivity matrix, denoted by if one substitutes vertex-degrees with the distance-sums (Szymanski et al., 1986) in the formula for the vertex-connectivity matrix, presented in Section 2.13  

The distance-sum-connectivity matrix is used for computing the weighted identification number (Szymanski et al., 1986), which has been successfully tested in QSAR (Bogdanov et al, 1987 Carter et al., 1987, 1988). Randid introduced the concept of identification numbers (Randic, 1984, 1986), while Szymanski et al. (1985, 1986) investigated its discriminatory power. They found that the identification numbers are highly discriminating indices, but they are not unique. For example, in the field of 618.050 trees representing all alkanes up to 20 carbon atoms, there are 175 pairs and 20 triplets of nonisomorphic structures with the same identification number. 

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The distance-sum-connectivity matrix analogous to the % matrix but based on the vertex distance sum a instead of the vertex degree 6 [Szymanski, Muller et al, 1986b] ... 

The distance-sum-connectivity matrix is used to calculate the —> Weighted ID number (WID). 

A variant of the path-% matrix is obtained by using the vertex distance sum a in place of the vertex degree as the vertex-weighting scheme. The matrix obtained in this way can be considered an extension of the —> distance-sum-connectivity matrix and thus called path-distance-sum-connectivity matrix, denoted by "Xy... 

The distance-sum-connectivity matrix of G, (see structure A in Figure 2.1) is a square 7x7 matrix, given below ... 

Distance-sum-connectivity matrix of 5-methyl-l,3,4-oxathiazol-2-one VS, is the matrix row sum. 

The formerly proposed and the most important of this series of topological indices is the Baiaban distance connectivity index J (also called distance connectivity index or average distance sum connectivity). It is one of the most discriminating - molecular descriptors and its values do not increase substantially with molecule size or number of rings it is defined in terms of sums over each ith row of the - distance matrix D, i.e. the vertex distance degree o [Baiaban, 1982 Baiaban, 1983a]. It is defined as ... 

Another index calculated from the distance matrix is the Balaban index / (or average distance sum connectivity index) [42], Each distance sum Dt is the sum of the elements of the th row of the distance matrix. The index is normalized by the numbers of bonds B and rings C, and is calculated from ... 

The sum-vertex-connectivity matrix, denoted by S, was introduced independently by Zhou and Trinajstid (2009, 2010a) and Randid et al. (2010). Randid et al. (2010) named this matrix the distance-weighted adjacency matrix. It is defined as follows ... 

Balaban proposed another connectivity index based on the distance matrix of molecular graph. He defined the sum of a column or a row of distance matrix as the distance sum of an atom. 

The minimal spanning tree also operates on the distance matrix. Here, near by patterns are connected with lines in such a way that the sum of the connecting lines is minimal and no closed loops are constructed. Here too the information on distances is retained, but the mutual orientation of patterns is omitted. Both methods, hierarchical clustering and minimal spanning tree, aim for making clusters in the multi-dimensional space visible on a plane. 

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... 

This matrix is a special case of —> distance degree matrices obtained by the parameter combination a = 0, (3 = 0, y=l. The row sum of the additive adjacency matrix is the —> extended connectivity of first-order EC defined by Morgan. This local invariant was used to calculate the eccentric adjacency index. A modification of this matrix, which accounts for heteroatoms, is the additive chemical adjacency matrix. 

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. 

The variable Balaban index, denoted as J, is calculated by analogy with the Balaban distance connectivity index from the row sums of the variable augmented distance matrix as [Randic and Pompe, 2001b]... 

Variable connectivity index, variable Balaban index, and variable Zagreb indices for 2-pentanol. A(x, y) and D(x, y) are the variable augmented adjacency matrix and the variable augmented distance matrix, respectively. VS, indicates the matrix row sums x and y are the variable parameters for carbon and oxygen atom, respectively. 

The Wiener index [86] can be expressed in terms of the distance matrix [87] and equals the half-sum of all distance matrix entries. Randi(5 [88] and Kier and Hall indices of order 0-3 [89] are calculated from coordination numbers of atoms or from values of atomic connectivity. The Kier shape index (order 1-3) [90] depends on the number of skeletal atoms, molecular branching, and the ratio of the atomic radius and the radius of the carbon atom in the sp hybridization state. The Kier flexibility index [90] is derived from the Kier shape index. The Balaban index depends on the row sums of the entries of the distance matrix and the cyclomatic number [92,93]. The information content index and its derivatives (order 0-2) are based on the Shannon information theory [95]. Modifications of the information content index are structural information content, complementary information content, and bond information content [96],... 

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