2. 17 The Augmented Vertex-Degree Matrix

The augmented vertex-degree matrix, denoted aΔ, is unsymmetric V × V matrix defined as [59-61]:

[aΔ]ij=
  d(j)/2l(i,j)         if i ≠ j
  d(i)                 if i = j                                 (36) 

where d(j) is the degree of the vertex j and l(i,j) the distance between vertices i and j.

As an illustrative example of the augmented vertex-degree matrix aΔ we give below this matrix for G1 (see structure A in Figure 2) whose vertex degrees are given in Figure 16.

aΔ(G1)=
1
2/2
3/4
2/8
2/16
3/8
1/16
1/2
2
3/2
2/4
2/8
3/4
1/8
1/4
2/2
3
2/2
2/4
3/2
1/4
1/8
2/4
3/2
2
2/2
3/4
1/8
1/16
2/8
3/4
2/2
2
3/2
1/4
1/8
2/4
3/2
2/4
2/2
3
2/2
1/16
2/8
3/4
2/8
2/4
3/2
1

The augmented vertex-degree matrix can be used to compute the complexity index proposed by Randić and Plavšić [59-61]. The Randić-Plavšić complexity index is equal to the sum of all the matrix row-sums for vertices non-equivalent by symmetry. It should be noted that the i-th row-sum represents the augmented degree of the vertex i.

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