The vertex-distance-path matrix, denoted by vDp, was introduced by Diudea [218,219]. Its entries are based on the elements of the vertex-distance matrix:
[vDp]ij=  |
|
||||
| 0 otherwise (46) |
It should be noted that the elements [vDp]ij count all internal paths included in the shortest paths between vertices i and j in a graph. An algorithm which finds all paths on a graph (molecular skeleton) has been devised by Randić et al. [220].
As examples of the vertex-distance-path matrices of (molecular) graphs, we give vDp matrices of T2 and G1 (see structure A in Figure 2):
vDp(T2)= |
 |
0 |
1 |
3 |
6 |
10 |
15 |
6 |
3 |
 |
1 |
0 |
1 |
3 |
6 |
10 |
3 |
1 |
|||
3 |
1 |
0 |
1 |
3 |
6 |
1 |
3 |
|||
6 |
3 |
1 |
0 |
1 |
3 |
3 |
6 |
|||
10 |
6 |
3 |
1 |
0 |
1 |
6 |
10 |
|||
15 |
10 |
6 |
3 |
1 |
0 |
10 |
15 |
|||
6 |
3 |
1 |
3 |
6 |
10 |
0 |
6 |
|||
3 |
1 |
3 |
6 |
10 |
15 |
6 |
0 |
vDp(G1)= |
|
0 |
1 |
3 |
6 |
10 |
6 |
10 |
|
1 |
0 |
1 |
3 |
6 |
3 |
6 |
|||
3 |
1 |
0 |
1 |
3 |
1 |
3 |
|||
6 |
3 |
1 |
0 |
1 |
3 |
6 |
|||
10 |
6 |
3 |
1 |
0 |
1 |
3 |
|||
6 |
3 |
1 |
3 |
1 |
0 |
1 |
|||
10 |
6 |
3 |
6 |
3 |
1 |
0 |
The vertex-distance-path matrix allows the direct computation of the hyper-Wiener index [219,377].
if i ≠ j 