5.16 The Quotient Matrices

Quotient matrices have been introduced by Randić [330] and applied to structure-property modeling by Nikolić et al. [261] and Plavšić et al. [331]. The quotient matrices, denoted by Ma/Mb, are obtained by dividing the off-diagonal elements of matrices Ma and Mb:

[Ma/Mb]ij=
  [Ma]ij/[Mb]ij          if i ≠ j
  0                           otherwise                             (118)    

Several quotient matrices are in use. Here we list six: the vertex-distance/detour matrix vD/DM [330] , the detour/vertex-distance matrix DM/vD [331], the vertex-distance/resistance-distance matrix vD/Ω [277,332], the resistance-distance/vertex-distance matrix Ω/vD [277,332], the vertex-distance/vertex-distance-complement matrix vD/vcD [261] and the vertex-distance-complement/vertex-distance matrix vcD/vD [261]. These six quotient matrices for G1 (see structure A in Figure 2) are given below.

vD/DM(G1)=
0
1
1
0.6
1
0.6
0.67
1
0
1
0.5
1
0.5
0.6
1
1
0
0.33
1
0.33
0.5
0.6
0.5
0.33
0
0.33
1
1
1
1
1
0.33
0
0.33
0.5
0.6
0.5
0.33
1
0.33
0
1
0.67
0.6
0.5
1
0.5
1
0

The molecular index based on the vertex-distance/detour matrix is called the Wiener-sum index [330] .

DM/vD(G1)=
0
1
1
1.67
1
1.67
1.5
1
0
1
2
1
2
1.67
1
1
0
3
1
3
2
1.67
2
3
0
3
1
1
1
1
1
3
0
3
2
1.67
2
3
1
3
0
1
1.5
1.67
2
1
1
1
0

The molecular index based on the detour/vertex-distance matrix is called the detour-sum index [331] .

vD/Ω(G1)=
0
1
1
1.09
1.33
1.09
1.07
1
0
1
1.14
1.5
1.14
1.09
1
1
0
1.33
2
1.33
1.14
1.09
1.14
1.33
0
1.33
2
1.5
1.33
1.5
2
1.33
0
1.33
1.14
1.09
1.14
1.33
2
1.33
0
1
1.07
1.09
1.14
1.5
1.14
1
0

The molecular index based on the vertex-distance/resistance-distance matrix is a variant of the Wiener-sum index [277] .

Ω/vD(G1)=
0
1
1
0.92
0.75
0.92
0.93
1
0
1
0.88
0.67
0.88
0.92
1
1
0
0.75
0.5
0.75
0.88
0.92
0.88
0.75
0
0.75
0.5
0.67
0.75
0.67
0.5
0.75
0
0.75
0.88
0.92
0.88
0.75
0.5
0.75
0
1
0.93
0.92
0.88
0.67
0.88
1
0

The molecular index based on the resistance-distance/vertex-distance matrix is called the Kirchhoff-sum index [277]. Matrices vD/W and W/vD have been used to study the graph cyclicity [332].

vD/vcD(G1)=
0
0.2
0.5
1
2
1
2
0.2
0
0.2
0.5
1
0.5
1
0.5
0.2
0
0.2
0.5
0.2
0.5
1
0.5
0.2
0
0.2
0.5
1
2
1
0.5
0.2
0
0.2
0.5
1
0.5
0.2
0.5
0.2
0
0.2
2
1
0.5
1
0.5
0.2
0

 

vcD/vD(G1)=
0
5
2
1
0.5
1
0.5
5
0
5
2
1
2
1
2
5
0
5
2
5
2
1
2
5
0
5
2
1
0.5
1
2
5
0
5
2
1
2
5
2
5
0
5
0.5
1
2
1
2
5
0

Two variants of the Balaban index [213] have been derived from the vD/vcD and vcD/vDmatrices [261] .

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