The Wiener matrix, also called the edge-Wiener matrix [22] and denoted by eW, was introduced for acyclic graphs [290]. It is a sparse symmetric square V × V matrix whose elements are defined as:
[eW]ij=  |
Vi,e Vj,e if i ≠ j |
| 0 otherwise (90) |
where Vi,e and Vj,e denote the number of vertices in the two subgraphs (fragments) after an edge i-j, denoted by e, is removed from the acyclic graph.
Below we give the edge-Wiener matrix for the branched tree T2 .
eW(T2)= |
 |
0 |
7 |
0 |
0 |
0 |
0 |
0 |
0 |
 |
7 |
0 |
15 |
0 |
0 |
0 |
0 |
7 |
|||
0 |
15 |
0 |
15 |
0 |
0 |
7 |
0 |
|||
0 |
0 |
15 |
0 |
12 |
0 |
0 |
0 |
|||
0 |
0 |
0 |
12 |
0 |
7 |
0 |
0 |
|||
0 |
0 |
0 |
0 |
7 |
0 |
0 |
0 |
|||
0 |
0 |
7 |
0 |
0 |
0 |
0 |
0 |
|||
0 |
7 |
0 |
0 |
0 |
0 |
0 |
0 |
Summation of non-zero elements in the upper or lower matrix-triangles gives the Wiener number. The edge-Wiener matrix was also found to be a rich and stimulating source of novel molecular descriptors [302,303] .
The sparse Wiener matrix can be made dense by considering paths instead of edges when building up the matrix elements [209]. This type of the Wiener matrix is called the path-Wiener matrix [22] and is denoted by pW. Its elements for a tree are defined as:
[pW]ij= |
Vi,p Vj,p if i ≠ j |
| 0 otherwise (91) |
where p denotes a path between vertices i and j. Here Vi,p and Vj,p represent the number of vertices on each side of path p, including vertices i and j.
The path-Wiener matrix for the branched tree T2 from Figure 24 is:
pW(T2)= |
|
0 |
7 |
5 |
3 |
2 |
1 |
1 |
1 |
|
7 |
0 |
15 |
9 |
6 |
3 |
3 |
7 |
|||
5 |
15 |
0 |
15 |
10 |
5 |
7 |
5 |
|||
3 |
9 |
15 |
0 |
12 |
6 |
3 |
3 |
|||
2 |
6 |
10 |
12 |
0 |
7 |
2 |
2 |
|||
1 |
3 |
5 |
6 |
7 |
0 |
1 |
1 |
|||
1 |
3 |
7 |
3 |
2 |
1 |
0 |
1 |
|||
1 |
7 |
5 |
3 |
2 |
1 |
1 |
0 |
Summation of elements in the upper or lower matrix-triangle gives the hyper-Wiener number [209,270,302] .
Diudea [218,219] has also defined the Wiener difference matrix, denoted by dW, as follows:
dW = pW - eW (92)
The non-zero elements of this matrix represent counts of paths larger than unity.
The Wiener difference matrix for T2 is exemplified below.
dW(T2)= |
|
0 |
0 |
5 |
3 |
2 |
1 |
1 |
1 |
|
0 |
0 |
0 |
9 |
6 |
3 |
3 |
0 |
|||
5 |
0 |
0 |
0 |
10 |
5 |
0 |
5 |
|||
3 |
9 |
0 |
0 |
0 |
6 |
3 |
3 |
|||
2 |
6 |
10 |
0 |
0 |
0 |
2 |
2 |
|||
1 |
3 |
5 |
6 |
0 |
0 |
1 |
1 |
|||
1 |
3 |
0 |
3 |
2 |
1 |
0 |
1 |
|||
1 |
0 |
5 |
3 |
2 |
1 |
1 |
0 |
