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  "body": "\n## Statement\n\nIf 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.\n\nAs 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.\n\nAnalogues of the Whitney isomorphism theorem have been proven for the line graphs of [[multigraphs|0C62PZXX02]], but are more complicated in this case.\n",
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