[avA]ij= 
Randić introduced in 1991 the variable vertex-connectivity index [40] using the concept of the augmented vertex-adjacency matrix [124,125]. Variants of adjacency matrices, called the augmented vertex-adjacency matrices and denoted byavA , are the vertex-adjacency matrices that possess non-zero values on the main diagonal:
[avA]ij= |
1 if the vertices i and j are adjacent |
| wii if i = j | |
| 0 otherwise (12) |
where wii is the weight at the vertex i.
Augmented vertex-adjacency matrices were introduced to be used for vertex-weighted graphs, that is, graphs with one or more of their vertices distinguished in some way from the rest of their vertices [2]. But the augmented vertex-adjacency matrices are also used via appropriate molecular descriptors to assess structural differences, such as, for example, the relative role of carbon atoms of acyclic and cyclic parts in alkylcycloalkanes [125].
An example of a vertex-weighted graph is shown in Figure 14. Below the figure, we give the augmented vertex-adjacency matrix of G4 , in which the set of vertices is split in two subsets: the first containing six vertices of one kind (denoted by x) and the second containing one vertex of a different kind (denoted by y).
Figure 14. A vertex-labeled vertex-weighted graph G4 representing the carbon skeleton of 2-ethyl-3methyl-1-azacyclobutane. A weighted vertex is denoted by the grey circle.
avA(G4)= |
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x |
1 |
0 |
0 |
0 |
0 |
0 |
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1 |
x |
1 |
0 |
0 |
0 |
0 |
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0 |
1 |
x |
1 |
0 |
1 |
0 |
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0 |
0 |
1 |
y |
1 |
0 |
0 |
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0 |
0 |
0 |
1 |
x |
1 |
0 |
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0 |
0 |
1 |
0 |
1 |
x |
1 |
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0 |
0 |
0 |
0 |
0 |
1 |
x |
Several topological indices have been used in their variable form, e.g., two formulations of the variable Zagreb indices [126,127], the variable vertex-connectivity index [40,124,125,128-130]. The variable Zagreb indices and the variable connectivity indices represent generalizations of the original indices introduced three decades ago [71,79].
G4 