ABSTRACT
The initial adjacency-plus-distance matrix, or vertex-Schultz matrix, denoted by vSM, has been introduced by Schultz (1989) and formalized by Müller et al. (1990). It is dened as the sum of the vertex-adjacency matrix and the vertex-distance matrix:
vSM = vA + vD (5.1)
An example of this matrix for the vertex-labeled graph G1 (see structure A in Figure 2.1) is as follows:
v GSM( )1
0 2 2 3 4 3 4
2 0 2 2 3 2 3
2 2 0 2 2 2 2
3 2 2 0 2 2 3
4 3 2 2 0 2 2
=
3 2 2 2 2 0 2
4 3 2 3 2 2 0
Schultz used the vertex-adjacency-plus-vertex-distance matrix vA + vD to derive what he called the molecular topological index (MTI) for alkanes by multiplying the row-matrix containing the vertex-degrees with vSM, and then summing up the elements of the obtained row-matrix (Schultz, 1989). Later on, the molecular topological index was named the Schultz index (Trinajstić, 1992; Mihalić and Trinajstić, 1992; Mihalić et al., 1992a), and this term is now generally accepted (e.g., Devillers and Balaban, 1999; Todeschini and Consonni, 2000, 2009; Diudea et al., 2001; Hong et al., 2012). The Schultz index has found moderate use in the structure-propertyactivity modeling (e.g., Mihalić and Trinajstić, 1992; Mihalić et al., 1992b; Jurić et al., 1992; Todeschini and Consonni, 2000, 2009). H.P. Schultz and his collaborators, E.B. Schultz and T.P. Schultz, published 12 papers on various aspects of the Schultz index (Todeschini and Consonni, 2009). It has also been shown that the Shultz index and the Wiener index are closely related graph-theoretical invariants for acyclic
structures (Klein et al., 1992; Plavšić et al., 1993; Gutman, 1994b; Gutman and Klavžar, 1997). A determinant of the vSM matrix has also been used as a molecular descriptor (Schultz et al., 1990; von Knop et al., 1991).
