On a conjecture about Wiener index in iterated line graphs of trees

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 ≥ 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.