5.8 Reciprocal Szeged Matrices

The reciprocal Szeged matrices, denoted by SZ-1, are matrices whose off-diagonal elements are the reciprocal of the corresponding elements of the Szeged matrices:

[SZ-1]ij = [SZ]ij-1                                   (99)    

All elements equal to zero in the Szeged matrices are left unchanged in the reciprocal Szeged matrices. Below we give the reciprocal Szeged matrices of the edge-Szeged matrix, the path-Szeged matrix and the Szeged difference matrix of G1 (see structure A in Figure 2).

eSZ-1(G1)=
0
1/6
0
0
0
0
0
1/6
0
1/10
0
0
0
0
0
1/10
0
1/10
0
1/12
0
0
0
1/10
0
1/12
0
0
0
0
0
1/12
0
10
0
0
0
1/12
0
1/10
0
1/6
0
0
0
0
0
1/6
0

 

pSZ-1(G1)=
0
1/6
1/5
1/10
1/8
1/10
1/6
1/6
0
1/10
1/4
1/12
1/6
1/12
1/5
1/10
0
1/10
1/3
1/12
1/4
1/10
1/4
1/10
0
1/12
1/2
1/10
1/8
1/12
1/3
1/12
0
1/10
1/2
1/10
1/6
1/12
1/2
1/10
0
1/6
1/6
1/12
1/4
1/10
1/2
1/6
0

 

dSZ-1(G1)=
0
0
1/5
1/10
1/8
1/10
1/6
0
0
0
1/4
1/12
1/6
1/12
1/5
0
0
0
1/3
0
1/4
1/10
1/4
0
0
0
1/2
1/10
1/8
1/12
1/3
0
0
0
1/2
1/10
1/6
0
1/2
0
0
0
1/6
1/12
1/4
1/10
1/2
0
0

These matrices have not been used so far to generate molecular descriptors.

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