information:classification, stylistic:tone, stylistic:style_transfer
Multigraphs

The concept of the line graph of G may naturally be extended to the case where G is a multigraph. In this case, the characterizations of these graphs can be simplified: the characterization in terms of clique partitions no longer needs to prevent two vertices from belonging to the same to cliques, and the characterization by forbidden graphs has seven forbidden graphs instead of nine.

However, for multigraphs, there are larger numbers of pairs of non-isomorphic graphs that have the same line graphs. For instance a complete bipartite graph K has the same line graph as the dipole graph and Shannon multigraph with the same number of edges. Nevertheless, analogues to Whitney's isomorphism theorem can still be derived in this case.