The modified edge-Wiener matrix, denoted by meW, is defined as [304]:
[meW]ij=  |
1/( Vi,e Vj,e ) if i ≠ j |
| 0 otherwise (93) |
Consequently, the modified edge-Wiener matrix meW can be directly obtained from the edge-Wiener matrix meW for the non-vanishing matrix elements:
[meW]ij = 1/ [eW]ij (94)
As an example, the modified Wiener matrix for the branched tree T2 (see Figure 24) is given below.
meW(T2)= |
 |
0 |
1/7 |
0 |
0 |
0 |
0 |
0 |
0 |
 |
1/7 |
0 |
1/15 |
0 |
0 |
0 |
0 |
1/7 |
|||
0 |
1/15 |
0 |
1/15 |
0 |
0 |
1/7 |
0 |
|||
0 |
0 |
1/15 |
0 |
1/12 |
0 |
0 |
0 |
|||
0 |
0 |
0 |
1/12 |
0 |
7 |
0 |
0 |
|||
0 |
0 |
0 |
0 |
7 |
0 |
0 |
0 |
|||
0 |
0 |
1/7 |
0 |
0 |
0 |
0 |
0 |
|||
0 |
1/7 |
0 |
0 |
0 |
0 |
0 |
0 |
Summation of non-zero elements in the upper or lower matrix-triangle gives the modified Wiener index. A generalization of the modified Wiener matrix leading to a class of modified Wiener indices has also been proposed [305] .
