1
2
3
4
5
6
7
8
1
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
1
1
0
0
0
1
0
0
0
0
0
1
0
0
1
0
0
0
1
0
The edge-vertex incidence matrix EV is an unsymmetrical E × V matrix which is the transpose of the vertex-edge incidence matrix VE. The EV matrix belonging to T2 is given below.
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
||||
| EV(T2)= | a | |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
0 |
|
| b | 0 |
1 |
1 |
0 |
0 |
0 |
0 |
0 |
|||
| c | 0 |
0 |
1 |
1 |
0 |
0 |
0 |
0 |
|||
| d | 0 |
0 |
0 |
1 |
1 |
0 |
0 |
0 |
|||
| e | 0 |
0 |
0 |
0 |
1 |
1 |
0 |
0 |
|||
| f | 0 |
1 |
0 |
0 |
0 |
0 |
0 |
1 |
|||
| g | 0 |
0 |
1 |
0 |
0 |
0 |
1 |
0 |
The incidence matrices VE and EV are related to the vertex-adjacency matrix vA of the graph G as [15]:
VE · EV = vA (G) + Δ (39)
where Δ is a diagonal matrix.
The matrices VE and EV are also related to the vertex-adjacency matrix vA of a line graph L(G) of G [12,15,19]:
EV · VE = vA (L(G)) - 2 I (40)
where I is unit V x V matrix.
When the incidence matrices VE and EV are associated with the oriented graph G, they are related to the Laplacian matrix L of G:
VE · EV = L (41)
