Line graph

In the line graph of an  G is another graph L(G) such that That is, it is the of the edges of G, representing each edge by the set of its two endpoints.
 * each of L(G) represents an edge of G; and
 * two vertices of L(G) are if and only if their corresponding edges share a common endpoint ("are incident") in G.

Example
The following figures show a graph (left, with blue vertices) and its line graph (right, with green vertices). Each vertex of the line graph is shown labeled with the pair of endpoints of the corresponding edge in the original graph. For instance, the green vertex on the right labeled 1,3 corresponds to the edge on the left between the blue vertices 1 and 3. Green vertex 1,3 is adjacent to three other green vertices: 1,4 and 1,2 (corresponding to edges sharing the endpoint 1 in the blue graph) and 4,3 (corresponding to an edge sharing the endpoint 3 in the blue graph).