svZ(T2)=
0
8
0
0
0
0
0
0
0
3
0
0
0
0
8
0
6
0
0
9
0
0
7
0
0
0
0
10
0
0
0
0
0
0
0
0
The first numerical matrix, using the Hosoya index, is called the edge-Hosoya matrix and is denoted by eZ. This matrix was already discussed in the section 5.12 where it was called simply the Hosoya matrix. If we sum the elements in one triangle of the eZ-matrix as originally suggested by Hosoya [25] when he defined the Wiener index from the distance matrix, the double invariant so obtained is called the edge-Hosoya-Wiener index.
The second numerical matrix is named the path-Hosoya matrix and denoted by pZ. This matrix is also discussed in the section 5.12 under the name the dense Hosoya matrix. If we sum up the elements in one triangle of the pZ-matrix, the index that is obtained is called the path-Hosoya-Wiener index.
The third numerical matrix is named the sparse vertex-Hosoya matrix and denoted by svZ. An example of this matrix obtained from the sparse vertex-graphical matrix of T2 is given below. Only the upper matrix-triangle is shown.
svZ(T2)= |
|
0 |
8 |
0 |
0 |
0 |
0 |
0 |
0 |
|
0 |
3 |
0 |
0 |
0 |
0 |
8 |
||||
0 |
6 |
0 |
0 |
9 |
0 |
|||||
0 |
7 |
0 |
0 |
0 |
||||||
0 |
10 |
0 |
0 |
|||||||
0 |
0 |
0 |
||||||||
0 |
0 |
|||||||||
0 |
If we sum up the elements in this triangle of the svZ-matrix, the resulting index is called the sparse vertex-Hosoya-Wiener index.
The fourth numerical matrix is named the dense vertex-Hosoyamatrix and denoted by dvZ. An example of this matrix obtained from the dense vertex-graphical matrix of T2 is given below. And again only the upper matrix-triangle is given.
dvZ(T2)= |
|
0 |
8 |
6 |
10 |
7 |
12 |
13 |
11 |
|
0 |
3 |
4 |
3 |
5 |
5 |
8 |
||||
0 |
6 |
3 |
6 |
9 |
6 |
|||||
0 |
7 |
7 |
8 |
10 |
||||||
0 |
10 |
7 |
7 |
|||||||
0 |
11 |
12 |
||||||||
0 |
13 |
|||||||||
0 |
If we sum up the elements in this triangle of the dvZ-matrix, the index that results is called the dense vertex-Hosoya-Wiener index.
