Line graph: Difference between revisions

Revision as of 04:59, 13 February 2023

Revision as of 07:45, 13 February 2023

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===Whitney isomorphism theorem===

[[File:Diamond line graph.svg|thumb|120px|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>