[δχ]ij=  |
[δ(i)δ(j)]-1/2 if vertices i and j are adjacent |
| 0 otherwise (88) |
where the distance-sum is defined as [215]:
δ(i)= |
V | [vD]ij |
(89) |
| ∑ | |||
| j=1 |
One can generate the distance-sum-connectivity matrix, denoted by δχ, if one substitutes vertex-degrees in the formula (14) for the vertex-connectivity matrix, presented in section 2.10, with the distance-sums [300]:
[δχ]ij= |
[δ(i)δ(j)]-1/2 if vertices i and j are adjacent |
| 0 otherwise (88) |
where the distance-sum is defined as [215]:
δ(i)= |
V | [vD]ij |
(89) |
| ∑ | |||
| j=1 |
For example, the distance-sums of vertices in G1 (see structure A in Figure 2), obtained from the corresponding vertex-distance matrix, given in section 4.1, are as follows (vertex-labels are in parenthesis): 17 (1), 12 (2), 9 (3), 12 (4), 13 (5), 10 (6) and 15 (7).
The distance-sum-connectivity matrix of G1 (see structure A in Figure 2) is a square 7 by 7 matrix, given below.
δχ(G1)= |
 |
0 |
0.070 |
0 |
0 |
0 |
0 |
0 |
 |
0.070 |
0 |
0.096 |
0 |
0 |
0 |
0 |
|||
0 |
0.096 |
0 |
0.096 |
0 |
0.105 |
0 |
|||
0 |
0 |
0.096 |
0 |
0.080 |
0 |
0 |
|||
0 |
0 |
0 |
0.080 |
0 |
0.088 |
0 |
|||
0 |
0 |
0.105 |
0 |
0.088 |
0 |
0.082 |
|||
0 |
0 |
0 |
0 |
0 |
0.082 |
0 |
The distance-sum-connectivity matrix is used for computing the weighted identification number [300], a number which has been successfully tested in QSAR [301].
