[vχ]ij= 
The vertex-connectivity matrix, denoted by vχ, was introduced by Randić [132]. It can be regarded as an edge-weighted matrix of a graph that is defined as:
[vχ]ij= |
[d(i)d(j)]-1/2 if vertices i and j are adjacent |
| 0 otherwise (14) |
where d(i) and d(j) are the degrees of vertices i and j.
For example, the degrees of vertices in G1 are given in Figure 16.
Figure 16. The vertex-degrees in G1.
The vertex-connectivety matrix of G1 (using the vertex-labels presented in structure A in Figure 2 and vertex-degrees from Figure 16) is given below.
| vχ(G1)= |  |
0 | 0.707 |
0 |
0 |
0 |
0 |
0 |
 |
| 0.707 | 0 |
0.408 |
0 |
0 |
0 |
0 |
|||
| 0 | 0.408 |
0 |
0.408 |
0 |
0.333 |
0 |
|||
| 0 | 0 |
0.408 |
0 |
0.500 |
0 |
0 |
|||
| 0 | 0 |
0 |
0.500 |
0 |
0.408 |
0 |
|||
| 0 | 0 |
0.333 |
0 |
0.408 |
0 |
0.577 |
|||
| 0 | 0 |
0 |
0 |
0 |
0.577 |
0 |
The summation of elements in the upper (or lower) matrix-triangle gives the vertex-connectivity index of G1 [370,375].
The vertex-connectivety matrix has also been used in computing the connectivity identification (ID) number [133,134]. The connectivity ID number was successfully tested in QSAR [135,136].
