---
tag: J47T08BQV3
kind: statement
role: theorem
name: Whitney isomorphism theorem
aliases: [whitney-graph-isomorphism-theorem]
sources:
  - wikipedia: en:675231@r1362295874
    locator: Line graph, section Properties > Whitney isomorphism theorem
    note: adopted
    mention: wikipedia://en/675231#D2C8
    sections: [Statement]
inventory: [P2NW0R2E3T]
---

## 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|0C62PZXX02]], but are more complicated in this case.