[vZM]ij= 
The Zagreb matrices, can also be considered as the vertex- and edge-weighted matrices related to the vertex- and edge-connectivity matrices. They can be formulated in terms of the vertex- or edge-degrees.
Zagreb matrices in terms of vertex-degrees
The vertex-Zagreb matrix, denoted by vZM, is a diagonal V × V matrix defined by:
[vZM]ij= |
[d(i)]2 if i =j |
| 0 otherwise (17) |
The vZM matrix for G1 (see structure A in Figure 2) is given below.
vZM(G1)= |
 |
1 | 0 |
0 |
0 |
0 |
0 |
0 |
 |
| 0 | 4 |
0 |
0 |
0 |
0 |
0 |
|||
| 0 | 0 |
9 |
0 |
0 |
0 |
0 |
|||
| 0 | 0 |
0 |
4 |
0 |
0 |
0 |
|||
| 0 | 0 |
0 |
0 |
4 |
0 |
0 |
|||
| 0 | 0 |
0 |
0 |
0 |
9 |
0 |
|||
| 0 | 0 |
0 |
0 |
0 |
0 |
1 |
Summation of the diagonal elements gives the first Zagreb index [71-74,76-78,109,367,376].
The modified vertex-Zagreb matrix, denoted by mvZM, is defined as:
[mvZM]ij= |
[d(i)]-2 if i =j |
| 0 otherwise (18) |
The mvZM matrix for G1 (see structure A in Figure 2) is as follows:
mvZM(G1)= |
|
1 | 0 |
0 |
0 |
0 |
0 |
0 |
|
| 0 | 1/4 |
0 |
0 |
0 |
0 |
0 |
|||
| 0 | 0 |
1/9 |
0 |
0 |
0 |
0 |
|||
| 0 | 0 |
0 |
1/4 |
0 |
0 |
0 |
|||
| 0 | 0 |
0 |
0 |
1/4 |
0 |
0 |
|||
| 0 | 0 |
0 |
0 |
0 |
1/9 |
0 |
|||
| 0 | 0 |
0 |
0 |
0 |
0 |
1 |
Summation of the diagonal elements gives the modified first Zagreb index [73,109,376].
The edge-Zagreb matrix, denoted by eZM, is defined by:
[eZM]ij= |
d(i)d(j) if vertices i and j are adjacent |
| 0 otherwise (19) |
The edge-Zagreb matrix eZM for G1 (see structure A in Figure 2) is presented below.
eZM(G1)= |
|
0 | 2 |
0 |
0 |
0 |
0 |
0 |
|
| 2 | 0 |
6 |
0 |
0 |
0 |
0 |
|||
| 0 | 6 |
0 |
6 |
0 |
9 |
0 |
|||
| 0 | 0 |
6 |
0 |
4 |
0 |
0 |
|||
| 0 | 0 |
0 |
4 |
0 |
6 |
0 |
|||
| 0 | 0 |
9 |
0 |
6 |
0 |
3 |
|||
| 0 | 0 |
0 |
0 |
0 |
3 |
0 |
Summation of the off-diagonal elements in the upper (or lower) matrix-triangle produces the second Zagreb index [71-73,75-78,109,376].
Finally, the modified edge-Zagreb matrix, denoted by meZM, is defined as:
[meZM]ij= |
[ d(i)d(j)]-1 if vertices i and j are adjacent |
| 0 otherwise (20) |
As an example, the modified edge-Zagreb matrix meZM for G1 (see structureA in Figure 2) is presented below.
| meZM(G1)= | |
0 | 1/2 |
0 |
0 |
0 |
0 |
0 |
|
| 1/2 | 0 |
1/6 |
0 |
0 |
0 |
0 |
|||
| 0 | 1/6 |
0 |
1/6 |
0 |
1/9 |
0 |
|||
| 0 | 0 |
1/6 |
0 |
1/4 |
0 |
0 |
|||
| 0 | 0 |
0 |
1/4 |
0 |
1/6 |
0 |
|||
| 0 | 0 |
1/9 |
0 |
1/6 |
0 |
1/3 |
|||
| 0 | 0 |
0 |
0 |
0 |
1/3 |
0 |
Summation of the off-diagonal elements in the upper (or lower) matrix-triangle produces the modified second Zagreb index [73,109,376].
Zagreb matrices in terms of edge-degrees
It should be noted that the Zagreb matrices of a graph G in terms of the edge-degrees are the vertex-Zagreb matrices of the corresponding line graph L(G). The edge-degrees in G1are given in Figure 17 and the vertex-degrees in L(G1), and the reader can easily confirm the above.
Zagreb indices found moderate use in structure property modeling [2,21,73,109,376]. In this respect, a contribution by Peng et al. [110] who showed how to improve the use of these indices in modeling molecular properties is important.
