Wiener index of iterated line graphs of trees homeomorphic to

Let G be a graph. Denote by L(G) its i-iterated line graph and denote by W (G) its Wiener index. Dobrynin, Entringer and Gutman stated the following problem: Does there exist a non-trivial tree T and i ≥ 3 such that W (L(T )) = W (T )? In a series of five papers we solve this problem. In a previous paper we proved that W (L(T )) > W (T ) for every tree T that is not homeomorphic to a path, claw K1,3 and to the graph of “letter H”, where i ≥ 3. Here we prove that W (L(T )) > W (T ) for every tree T homeomorphic to the claw, T 6= K1,3 and i ≥ 4.