G3Weighted graphs in chemistry usually represent heterosystems [113-116]. Molecules containing heteroatoms and heterobonds are represented by the vertex- and edge-weighted graphs [2]. A vertex- and edge-weighted graph Gvew is a graph which has one or more of its vertices and edges distinguished in some way from other vertices and edges in Gvew. These 'different' vertices and edges are weighted - their weights are usually identified by parameters h and k, respectively. In Figure 12, we give a vertex- and edge-weighted graph G3 corresponding, for example, to 2,6-diazanaphthalene.
Figure 12. A vertex- and edge-weighted graph G3 representing 2,6-diazanaphthalene.
The vertex-adjacency matrix of the vertex- and edge-weighted graph vA(Gvew) is defined by:
[vA(Gvew)]ij=  |
k if the edge i - j is weighted |
| 1 if the edge i - j is not weighted | |
| h if i = j and if the vertex i is weighted | |
| 0 otherwise (10) |
The parameters h and k depend, respectively, on the chemical nature of the corresponding atoms and bonds in a molecule. Some people select for them the values of the Hückel parameters for heteroatoms and heterobonds. Below is given the vertex-adjacency matrix for G3 from Figure 12.
vA(G3)= |
 |
0 |
1 |
0 |
0 |
0 |
0 |
0 | 0 | 0 |
k |  |
1 |
0 |
1 |
0 |
0 |
0 | 0 | 0 | 0 | 0 | |||
0 |
1 |
0 |
1 |
0 |
0 | 0 | 1 | 0 | 0 | |||
0 |
0 |
1 |
0 |
k |
0 | 0 | 0 | 0 | 0 | |||
0 |
0 |
0 |
k |
h |
k | 0 | 0 | 0 | 0 | |||
0 |
0 |
0 |
0 |
k |
0 | 1 | 0 | 0 | 0 | |||
0 |
0 |
0 |
0 |
0 |
1 | 0 | 1 | 0 | 0 | |||
0 |
0 |
1 |
0 |
0 |
0 | 1 | 0 | 1 | 0 | |||
0 |
0 |
0 |
0 |
0 |
0 | 0 | 1 | 0 | k | |||
k |
0 |
0 |
0 |
0 |
0 |
0 | 0 | k |
h |
