Adjacency-Plus-Distance Matrices

The initial adjacency-plus-distance matrix, or vertex-Schultz matrix, denoted by SM, has been introduced by Schultz (1989) and formalized by Muller et al. (1990). It is defined as the sum of the vertex-adjacency matrix and the vertex-distance matrix  

From (5.1), it can easily be seen that various versions of the adjacency matrix and distance matrix may be employed to generate other kinds of SM matrices and the corresponding versions of Schultz indices. For example, Estrada and Gutman (1996) introduced the edge-adjacency-plus-edge-distance matrix of a graph G  

This matrix is equal to the vertex-adjacency-plus-vertex-distance matrix of the respective line graph L(G). 1 is as follows  

Estrada and Gutman (1996) used this matrix to derive the edge-molecular topological index that we call the edge-Schultz index. 

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Knop, J. V., et al., On the Determinant of the Adjacency-Plus-Distance Matrix as the Topological Index for Characterizing Alkanes. J. Chem. Inf. Comput. Sci., 1991 31,83-84. 

The most popular sum matrices are the adjacency-plus-distance matrix obtained by summing the vertex adjacency matrix A and the vertex distance matrix D and the —> edge-adjacency-plus-edge-distance matrix ohtainedhy summing the edge adjacency matrix... 

The determinant of the 3D adjacency-plus-geometry matrix ( A + G) was also proposed as a molecular topographic descriptor A is the 3D adjacency matrix or bond length-weighted adjacency matrix whose entries corresponding to bonded atoms are bond distances instead of 1 and G the geometry matrix [Mihalic, Nikolic et al, 1992]. 

It is the largest eigenvalue of the —> a acency-plus-distance matrix obtained as the sum of the adjacency matrix A and the topological distance matrix D representing a molecular graph [Schultz, Schultz et al, 1990] = MaxSp A + D). Its logarithm was used to model physico-... 

The matrix A + is called edge-adjacency-plus-edge-distance matrix and denoted as... 

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