[vDvd (p,q,r)]ij=  |
l p(i,j)d q(i)d r(j) if i ≠ j |
| 0 otherwise (81) |
where, as before, l(i,j) is the shortest graph-theoretical distance between vertices i and j, and d(i), d(j) are the degrees of vertices i and j. The parameters p,q and r are natural numbers. The structure of a particular vertex-distance-vertex-degree matrix depends on the selected numerical values of these parameters. For example, if we choose the following values for the parameters: p=1, q=0 and r=0, then the vDvd matrix obtained will be identical to the vertex-distance matrix. If, on the other hand, we choose a slightly different set of parameter values such as p=-1, q=0 and r=0, then the vDvd matrix obtained is identical with the vertex-Harary matrix. As an example of computing the vertex-distance-vertex-degree matrix, we give below this matrix for T2 with parameters p=1, q=1 and r=1.
vDvd(1,1,1,T2)= |
 |
0 |
3 |
6 |
6 |
8 |
5 |
3 |
2 |
 |
3 |
0 |
9 |
12 |
18 |
12 |
6 |
3 |
|||
6 |
9 |
0 |
6 |
12 |
9 |
3 |
6 |
|||
6 |
12 |
6 |
0 |
4 |
4 |
4 |
6 |
|||
8 |
18 |
12 |
4 |
0 |
2 |
6 |
8 |
|||
5 |
12 |
9 |
4 |
2 |
0 |
4 |
5 |
|||
3 |
6 |
3 |
4 |
6 |
4 |
0 |
3 |
|||
2 |
3 |
6 |
6 |
8 |
5 |
3 |
0 |
From the definition of vertex-distance-vertex-degree matrix, it can be seen that non-symmetric vDvd matrices are obtained if q≠ r. As an example consider G1 (see structure A in Figure 2) with parameters: p=1, q=2 and r=1.
