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 gave a problem if there exists a nontrivial tree T and i≥3, such that W(Lⁱ(T))=W(T). As the key ingredient in the solution of this problem we prove here that for fixed graph G (which is not necessarily a tree), the function W(Lⁱ(G)) is convex in variable i. In this paper we also present a direct application of the convexity of W(Lⁱ(G)) to prove the above mentioned problem for trees without pendant paths of length greater than 1.