Abstract. Let G be a graph. Denote by Li(G) its i-iterated line graph and denote by W(G) its Wiener index. Dobrynin, Entringer and Gutman formulated a problem of characterizing all nontrivial trees T and i≥3, such that W(Li(T))=W(T).

In a series of papers we completely solve this problem. This is 4th paper of the series and we prove here that there is no tree T homeomorphic to H and i≥4 such that W(Li(T))=W(T). As a consequence we obtain that, if W(Li(T))=W(T) is true for some tree T and i≥3, then i=3.