The Hosoya matrix was introduced by Randić [316]. Denoted by Z, the Hosoya matrix, is derived here in a manner similar to the edge-Wiener matrix (see section 5.3). It is a sparse symmetric square V x V matrix whose elements for a tree are defined as:
[Z]ij=  |
Zi Zj if vertices i and j are adjacent |
0 otherwise (109) |
where Zi and Zj are the Hosoya Z-indices [25] of the two subgraphs (fragments) after an edge i-j is removed from the acyclic graph. The Z-indices are tabulated for C1 to C9 fragments [93] and programs for computing this and many other molecular descriptors are available [ e.g ., 317].
Z(T2)= |
 |
0 |
19 |
0 |
0 |
0 |
0 |
0 |
0 |
 |
19 |
0 |
24 |
0 |
0 |
0 |
0 |
19 |
|||
0 |
24 |
0 |
21 |
0 |
0 |
18 |
0 |
|||
0 |
0 |
21 |
0 |
20 |
0 |
0 |
0 |
|||
0 |
0 |
0 |
20 |
0 |
17 |
0 |
0 |
|||
0 |
0 |
0 |
0 |
17 |
0 |
0 |
0 |
|||
0 |
0 |
18 |
0 |
0 |
0 |
0 |
0 |
|||
0 |
19 |
0 |
0 |
0 |
0 |
0 |
0 |
The Hosoya matrix may be made dense if the elements [Z]ij are computed not only for deleted edges but also for deleted edges along any path in a tree [316] . Then, the dense Hosoya matrix dZ is defined as:
[dZ]ij= |
Zi Zj if i ≠ j |
| 0 otherwise (110) |
The dense Hosoya matrix for the branched tree T2 is exemplified below.
dZ(T2)= |
|
0 |
19 |
16 |
12 |
8 |
4 |
10 |
11 |
|
19 |
0 |
24 |
18 |
12 |
6 |
15 |
19 |
|||
16 |
24 |
0 |
21 |
14 |
7 |
18 |
16 |
|||
12 |
18 |
21 |
0 |
20 |
10 |
12 |
12 |
|||
8 |
12 |
14 |
20 |
0 |
17 |
8 |
4 |
|||
4 |
6 |
7 |
10 |
17 |
0 |
4 |
4 |
|||
10 |
15 |
18 |
12 |
8 |
4 |
0 |
10 |
|||
11 |
19 |
16 |
12 |
4 |
4 |
10 |
0 |
The Hosoya matrices are used to produce a variety of molecular descriptors, especially since the Z-index and the Hosoya matrix have been extended to polycyclic systems and edge-weighted graphs [ e.g ., 318-320] .
