eD(G1)=
0
1
2
3
3
3
2
1
0
1
2
2
2
1
2
1
0
1
2
2
1
3
2
1
0
1
2
2
3
2
2
1
0
1
1
3
2
2
2
1
0
1
2
1
1
2
1
1
0
The edge-distance matrix of a graph G, denoted by eD, is the vertex-distance matrix of the corresponding line graph L(G):
eD(G) = vD[L(G)] (50)
This is because the edge-distances in a graph are equal to the distances between vertices in the corresponding line graph. We give below the edge-distance matrix of G1 (see structure B in Figure 2) as the vertex-distance matrix of L(G1).
eD(G1)= |
|
0 |
1 |
2 |
3 |
3 |
3 |
2 |
|
1 |
0 |
1 |
2 |
2 |
2 |
1 |
|||
2 |
1 |
0 |
1 |
2 |
2 |
1 |
|||
3 |
2 |
1 |
0 |
1 |
2 |
2 |
|||
3 |
2 |
2 |
1 |
0 |
1 |
1 |
|||
3 |
2 |
2 |
2 |
1 |
0 |
1 |
|||
2 |
1 |
1 |
2 |
1 |
1 |
0 |
The edge-distance matrix of a graph G can also be simply constructed without considering the related line graph by counting vertices between the edges of G:
[eD]ij=  |
nij + 2 if i ≠ j |
| 0 otherwise (51) |
where nij is the number of vertices on the shortest path between edges i and j.
