Wiener index of iterated line graphs of trees homeomorphic to H

This is fourth paper out of five in which we completely solve a problem of Dobrynin, Entringer and Gutman. Let G be a graph. Denote by Li(G) its i-iterated line graph and denote by W(G) its Wiener index. Moreover, denote by H a tree on six vertices, out of which two have degree 3 and four have degree 1. Let j≥3. In previous papers we proved that for every tree T, which is not homeomorphic to a path, claw K1,3 and H, it holds W(Lj(T))>W(T). Here we prove that W(L4(T))>W(T) for every tree T homeomorphic to H. As a consequence, we obtain that with the exception of paths and the claw K1,3, for every tree T it holds W(Li(T))>W(T) whenever i≥4.