[vД]ij=  |
Nij[D]ij if i ≠ j |
| 0 otherwise (114) |
where Nij is the number of all shortest paths containing the path p(i,j) as a subpath.
The expanded vertex-distance matrix, denoted by vД, has been introduced by Tratch et al. [327]. It is a square symmetric V × V matrix defined as [24]:
[vД]ij= |
Nij[D]ij if i ≠ j |
| 0 otherwise (114) |
where Nij is the number of all shortest paths containing the path p(i,j) as a subpath.
The matrix vД for graph G1 (see structure A in Figure 2) is given by:
vД(G1)= |
 |
0 |
7 |
12 |
6 |
4 |
9 |
4 |
 |
7 |
0 |
12 |
8 |
12 |
12 |
6 |
|||
12 |
12 |
0 |
6 |
12 |
9 |
6 |
|||
6 |
8 |
6 |
0 |
4 |
8 |
6 |
|||
4 |
12 |
12 |
4 |
0 |
5 |
4 |
|||
9 |
12 |
9 |
8 |
5 |
0 |
7 |
|||
4 |
6 |
6 |
6 |
4 |
7 |
0 |
For acyclic graphs, the expanded vertex-distance matrix is given by:
[vД]ij= |
l(i,j) · Vi,p· Vj,p if i ≠ j |
| 0 otherwise (115) |
or in other words, the elements of the matrix vД for acyclic graphs can be simply obtained from the elements of vD and pW by the Hadamard matrix product:
vД = vD • pW (116)
The expanded vertex-distance matrix of T2 from Figure 24 is as follows:
vД(T2)= |
|
0 |
7 |
10 |
9 |
8 |
5 |
3 |
2 |
|
7 |
0 |
15 |
18 |
18 |
12 |
6 |
7 |
|||
10 |
15 |
0 |
15 |
20 |
15 |
7 |
10 |
|||
9 |
18 |
15 |
0 |
12 |
12 |
6 |
9 |
|||
8 |
18 |
20 |
12 |
0 |
7 |
6 |
8 |
|||
5 |
12 |
15 |
12 |
7 |
0 |
4 |
5 |
|||
3 |
6 |
7 |
6 |
6 |
4 |
0 |
3 |
|||
2 |
17 |
10 |
9 |
8 |
5 |
3 |
0 |
If the path-Wiener matrix is substituted by the unsymmetric Cluj matrix, then the unsymmetric expanded vertex-distance matrix is obtained. This has been used for a new definition of the hyper-Wiener index [314,328]. This kind of expanded vertex-distance matrix of T2 is shown below.
vД(T2)= |
|
0 |
1 |
2 |
3 |
4 |
5 |
3 |
2 |
|
7 |
0 |
3 |
6 |
9 |
12 |
6 |
7 |
|||
10 |
5 |
0 |
5 |
10 |
15 |
7 |
10 |
|||
9 |
6 |
3 |
0 |
6 |
12 |
6 |
9 |
|||
8 |
6 |
4 |
2 |
0 |
7 |
6 |
8 |
|||
5 |
4 |
3 |
2 |
1 |
0 |
4 |
5 |
|||
3 |
2 |
1 |
2 |
3 |
4 |
0 |
3 |
|||
2 |
1 |
2 |
3 |
4 |
5 |
3 |
0 |
Diudea and Gutman [314] generalized the concept of the expanded vertex-distance matrix in order to define novel matrices using the Hadamard matrix product between the vertex-distance matrix vD and a general square V × V matrix Ψ as:
vД Ψ= vD • Ψ (117)
If Ψ is a Szeged matrix, for example, the expanded vertex-distance Szeged matrix is derived. Todeschini and Consonni in their Handbook of Molecular Descriptors [21] listed 28 expanded vertex-distance matrices (see Table E-8 in that book). From these matrices, two kinds of molecular descriptors are derived, i.e., expanded distance indices and expanded square distance indices [329].
