Whitney isomorphism theorem

Statement

If the line graphs of two connected graphs are isomorphic, then the underlying graphs are isomorphic, except in the case of the triangle graph $K_{3}$ and the claw $K_{1,3}$, which have isomorphic line graphs but are not themselves isomorphic.

As well as $K_{3}$ and $K_{1,3}$, there are some other exceptional small graphs with the property that their line graph has a higher degree of symmetry than the graph itself. For instance, the diamond graph $K_{1,1,2}$ (two triangles sharing an edge) has four graph automorphisms but its line graph $K_{1,2,2}$ has eight. In the illustration of the diamond graph shown, rotating the graph by 90 degrees is not a symmetry of the graph, but is a symmetry of its line graph. However, all such exceptional cases have at most four vertices. A strengthened version of the Whitney isomorphism theorem states that, for connected graphs with more than four vertices, there is a one-to-one correspondence between isomorphisms of the graphs and isomorphisms of their line graphs.

Analogues of the Whitney isomorphism theorem have been proven for the line graphs of multigraphs, but are more complicated in this case.