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On a conjecture about Wiener index in iterated line graphs of trees
Knor, Martin ; Potočnik, Primož, 1971- ; Škrekovski, Riste
Let ▫$G$▫ be a graph. Denote by ▫$L^i(G)$▫ its ▫$i$▫-iterated line graph and denote by ▫$W(G)$▫ its Wiener index. Dobrynin and Melnikov conjectured that there exists no nontrivial tree ▫$T$▫ and ...▫$i\ge 3$▫, such that ▫$W(L^i(T)) = W(T)$▫. We prove this conjecture for trees which are not homeomorphic to the claw ▫$K_{1,3}$▫ and ▫$H$▫, where ▫$H$▫ is a tree consisting of 6 vertices, 2 of which have degree 3.
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