[aD]ij=  |
l(i,j) if i ≠ j |
| δij if i = j (53) |
where δij is the variable of weight of a vertex i.
The augmented vertex-distance matrix [224] of a vertex-labeled connected vertex-weighted graph vwG, denoted by aD, is a real symmetric V × V matrix whose elements are defined as:
[aD]ij= |
l(i,j) if i ≠ j |
| δij if i = j (53) |
where δij is the variable of weight of a vertex i.
The augmented distance matrix of the vertex-labeled weighted graph G4 in Figure 14) is given by:
aD(G4)= |
 |
δx |
1 |
2 |
3 |
4 |
3 |
4 |
 |
1 |
δx |
1 |
2 |
3 |
2 |
3 |
|||
2 |
1 |
δx |
1 |
2 |
1 |
2 |
|||
3 |
2 |
1 |
δy |
1 |
2 |
3 |
|||
4 |
3 |
2 |
1 |
δx |
1 |
2 |
|||
3 |
2 |
1 |
2 |
1 |
δx |
1 |
|||
4 |
3 |
2 |
3 |
2 |
1 |
δx |
Here, δx and δy are variable parameters, which are determined during the regression so that the standard error of estimate for a studied property is as small as possible.The augmented distance matrix has been used for generation of a number of variable distance indices [224,225]: the variableWiener index, the variable hyper-Wiener index, the variable Balaban index, and variable complements of these indices based on augmented distance-complement matrix .
