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==Iterating the line graph operator== {{harvtxt|van Rooij|Wilf|1965}} consider the sequence of graphs :<math>G, L(G), L(L(G)), L(L(L(G))), \dots.\ </math> 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}}}}, 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>
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