Wiener index of iterated line graphs of trees homeomorphic to

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 L i ( 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 K 1 , 3 and H , it holds W ( L j ( T ) ) > W ( T ) . Here we prove that W ( L 4 ( T ) ) > W ( T ) for every tree T homeomorphic to H . As a consequence, we obtain that with the exception of paths and the claw K 1 , 3 , for every tree T it holds W ( L i ( T ) ) > W ( T ) whenever i ≥ 4 .