[EvA]ij= 
The extended vertex-adjacency matrix, denoted by EvA, is a square symmetric V × V matrix defined as [137]:
[EvA]ij= |
([d(i)/d(j)]+[d(j)/d(i)])/2 if vertices i and j are adjacent |
| 0 otherwise (15) |
where d(i) and d(j) are the degrees of vertices i and j, respectively. This definition indicates that EvA matrix is a sort of edge-weighted vertex-adjacency matrix.
The extended vertex-adjacency matrix of G1 (see structure A in Figure 2) is presented as follows. The vertex-degrees in G1 are given in Figure 16.
| EvA(G1)= |  |
0 |
1.25 |
0 |
0 |
0 |
0 |
0 |
 |
1.25 |
0 |
1.08 |
0 |
0 |
0 |
0 |
|||
0 |
1.08 |
0 |
1.08 |
0 |
1.00 |
0 |
|||
0 |
0 |
1.08 |
0 |
1.00 |
0 |
0 |
|||
0 |
0 |
0 |
1.00 |
0 |
1.08 |
0 |
|||
0 |
0 |
1.00 |
0 |
1.08 |
0 |
1.67 |
|||
0 |
0 |
0 |
0 |
0 |
1.67 |
0 |
The use of topological indices based on this matrix in QSPR is explored by Yang et al. [137]. However, these authors did not consider the extended edge-adjacency matrix, denoted by EeA. The EeA matrix is based on the edge-degrees. Since the edge-degrees of a graph G are equal to vertex-degrees of a line graph L(G), it follows that:
EeA(G) = EvA(L(G)) (16)
From equation (16) is also evident that the extended edge-adjacency matrix is also a sort of edge-weighted adjacency matrix.
The extended edge-adjacency matrix of G1is equal to the extended vertex-adjacency matrix of L(G1).
| EeA(G1)=EvA[L(G1)]= | |
0 |
1.25 |
0 |
0 |
0 |
0 |
0 |
|
1.25 |
0 |
1.00 |
0 |
0 |
0 |
1.04 |
|||
0 |
1.00 |
0 |
1.08 |
0 |
0 |
1.04 |
|||
0 |
0 |
1.08 |
0 |
1.08 |
0 |
0 |
|||
0 |
0 |
0 |
1.08 |
0 |
1.08 |
1.04 |
|||
0 |
0 |
0 |
0 |
1.08 |
0 |
1.25 |
|||
0 |
1.04 |
1.04 |
0 |
1.04 |
1.25 |
0 |
Topological indices based on the extended edge-adjacency matrix have not yet been explored in QSPR or QSAR modeling.
