They show that, when ''{{mvar|G''}} is a finite [[connected graph]], only four behaviors are possible for this sequence:
*If ''{{mvar|G''}} is a [[cycle graph]] then {{math|''L''(''G'')}} and each subsequent graph in this sequence are [[graph isomorphism|isomorphic]] to ''{{mvar|G''}} itself. These are the only connected graphs for which {{math|''L''(''G'')}} is isomorphic to ''{{mvar|G''}}.<ref>This result is also Theorem 8.2 of {{harvtxt|Harary|1972}}.</ref>
*If ''{{mvar|G''}} is a claw {{math|''K''<{{sub>|1,3</sub>}}}}, then {{math|''L''(''G'')}} and all subsequent graphs in the sequence are triangles.
*If ''{{mvar|G''}} is a [[path graph]] then each subsequent graph in the sequence is a shorter path until eventually the sequence terminates with an [[empty graph]].
*In all remaining cases, the sizes of the graphs in this sequence eventually increase without bound.
If ''{{mvar|G''}} is not connected, this classification applies separately to each component of ''{{mvar|G''}}.
For connected graphs that are not paths, all sufficiently high numbers of iteration of the line graph operation produce graphs that are Hamiltonian.<ref>{{harvtxt|Harary|1972}}, Theorem 8.11, p. 81. Harary credits this result to [[Gary Chartrand]].</ref>
