The Augmented Vertex-Adjacency Matrix

Randic introduced in 1991 the variable vertex-connectivity index (Randic, 1991a) using the concept of the augmented vertex-adjacency matrix (Randic, 1991a, 1991b Randic et al., 2001). Variants of adjacency matrices, called the augmented vertex-adjacency matrices and denoted by A, are the vertex-adjacency matrices that possess nonzero values on the main diagonal  

FIGURE 2.15 A vertex-labeled vertex-weighted graph representing the carbon skeleton 

CgN of 2-ethyl-3-methyl-l-azacyclobutane Cf NH,3. The weighted vertex is denoted hy the dark circle. 

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 (Trinajstid, 1983,1992). 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 (Randid et al., 2001). 

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