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.