Editing Line graph (section)

===Whitney isomorphism theorem===
[[File:Diamond line graph.svg|thumb|The [[diamond graph]] (left) and its more-symmetric line graph (right), an exception to the strong Whitney theorem]]
If the line graphs of two [[connected graph]]s are isomorphic, then the underlying graphs are isomorphic, except in the case of the triangle graph {{math|''K''{{sub|3}}}} and the [[Claw (graph theory)|claw]] {{math|''K''{{sub|1,3}}}}, which have isomorphic line graphs but are not themselves isomorphic.<ref name="whitney">{{harvtxt|Whitney|1932}}; {{harvtxt|Krausz|1943}}; {{harvtxt|Harary|1972}}, Theorem 8.3, p.&nbsp;72. Harary gives a simplified proof of this theorem by {{harvtxt|Jung|1966}}.</ref>

As well as {{math|''K''{{sub|3}}}} and {{math|''K''{{sub|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]] {{math|''K''{{sub|1,1,2}}}} (two triangles sharing an edge) has four [[graph automorphism]]s but its line graph {{math|''K''{{sub|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.<ref>{{harvtxt|Jung|1966}}; {{harvtxt|Degiorgi|Simon|1995}}.</ref>

Analogues of the Whitney isomorphism theorem have been proven for the line graphs of [[multigraph]]s, but are more complicated in this case.<ref name="z97"/>