[ewA]ij= 
The edge-weighted edge-adjacency matrix, denoted by ewA, has been introduced by Estrada [131]. It is is a square unsymmetric E × E matrix defined as:
[ewA]ij= |
1 if edges i and j are adjacent |
| k if the edge i - j is weighted | |
| 0 otherwise (13) |
As an example of the edge-weighted edge-adjacency matrix, we present this matrix for the edge-weighted graph G5 depicted in Figure 15.
Figure 15. Edge-weighted graph G5.
The ewA matrix of G5 is as follows:
ewA(G5)= |
 |
0 |
1 |
0 |
0 |
0 |
0 |
0 |
 |
1 |
0 |
k |
0 |
0 |
0 |
1 |
|||
0 |
1 |
0 |
k |
0 |
0 |
1 |
|||
0 |
0 |
k |
0 |
1 |
0 |
0 |
|||
0 |
0 |
0 |
k |
0 |
1 |
1 |
|||
0 |
0 |
0 |
0 |
1 |
0 |
1 |
|||
0 |
1 |
k |
0 |
1 |
1 |
0 |
The parameter k is the edge-parameter (the bond-parameter) and Estrada presented in his paper [131] the values of this parameter for most of the bonds common in organic compounds. Estrada used the molecular descriptor, the edge-connectivety index, based on the edge-weighted edge-adjacency matrix to predict successfully the molecular volumes of 112 aliphatic organic compounds [374].
G5