Distance-Degree Matrices

There are two kinds of the distance-degree matrices one is based on the vertex-distance matrix and the vertex-degrees, and the other is based on the edge-distance matrix and edge-degrees. 

The vertex-distance-vertex-degree matrix of a simple graph G with V vertices, denoted by D d (p, q, r), is a square VxVmatrix defined as follows (Ivanciuc, 1999, 2000c)  

From the definition of vertex-distance-vertex-degree matrix, it can be seen that nonsymmetric D d matrices are obtained if r. As an example, consider G, (see 

The distance-degree matrices can be used to generate the distance-degree descriptors (Ivanciuc, 1989 Janezid et al., 2007 Gutman, 2013). 

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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. 

Some distance-degree matrices, representing simple molecular graphs and derived from selected combinations of a, P, and y parameters, result into other well-known graph-theoretical matrices defined in the literature. Examples are the —> distance matrix (a = 1, P = 0, y = 0), the —> Harary matrix (a = —l, P = 0, y = 0), and —> XI matrix (a = 0, P=—1/2, y=—1/2). 

Opposite to the distance matrix is the —> detour matrix, where the entries correspond to the length of the longest path between two vertices. Other matrices related to the distance matrix are geometric distance/topolo cal distance quotient matrix, —> detour-distance combined matrix, distance/detour quotient matrix, —> distance-degree matrices, —> expanded distance matrices, —> distance-path matrix, delta matrix, —> distance sum layer matrix, —> distance-sequence matrix. 

Other LOVIs similar to VTI indices are derived as the row sums of distance-degree matrices for different combinations of (3, y, and ([) parameters. These local indices were extensively... 

Other classes of graph-theoretical matrices are walk matrices, layer matrices, distance-degree matrices, and matrices from which Schultz-type indices are derived. 

Note. The vertex-distance-vertex-degree matrices with P = 0 were called v d matrices by Perdih [Perdih and Perdih, 2002a] and the general distance-degree matrix was denoted by G a,b,c) [Perdih and Perdih, 2004], whose elements are v vj dl [ Perdih and Perdih, 2003c], where v denotes the vertex degree 6 and a, b, and c are the parameters corresponding to p, y, and a, respectively. 

The diagonal elements of these matrices are equal to zero. Using the notations adopted in this book, the general distance-degree matrix elements are defined as the following ... 

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 ... 

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 maximum/minimum path sum of the i th vertex, denoted by MmPVS, is a local vertex invariant defined as the sum of the lengths of the longest and shortest paths between vertex v, and any other vertex in the molecular graph. It is calculated as the sum of elements over the / th row and / th column in the A/D matrix, or, alternatively, as the sum of the - vertex distance degree o, calculated on the distance matrix D and the maximum path sum MPVS, of the / th vertex calculated on the detour matrix A ... 

The row sums of this matrix contain information on the molecular folding in fact, in highly folded structures, they tend to be relatively small as the interatomic distances are small while the topological distances increase as the size of the structure increases. Therefore, the average row sum is a molecular invariant called the average distance/ distance degree, i.e. 

The vertex distance degree (or distance number, distance index, distance rank, vei tex distance sum, distance of a vertex) is the row sum a, of the distance matrix D ... 

The average row sum of the distance matrix is a molecular invariant called the average distance degree defined as ... 

The sum of the edge distance degrees, i.e. the sum of all matrix elements, is called total edge distance De and defined as ... 

A layer matrix obtained weighting the atoms by their -> vertex distance degree o, i.e. the row sum of the -> distance matrix D [Balaban and Diudea, 1993]. Therefore, the entry i-k of the layer matrix is the sum of the vertex distance degrees of the atoms located at distance k from the focused ith vertex. It is obvious that the entries of the first column k = 0) are only vertex distance degrees. Moreover, the sums over each row in LDS are all equal to twice the -> Wiener index, i.e. the following relation holds ... 

The Balaban DJ index was still defined in terms of modified vertex distance degrees a but using the formula of the —> matrix sum indices as JBalaban and Diudea, 1993]... 

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