The Wiener index in iterated line graphs

For a graph GG, denote by Li(G)Li(G) its ii-iterated line graph and denote by W(G)W(G) its Wiener index. We prove that the function W(Li(G))W(Li(G)) is convex in variable ii. Moreover, this function is strictly convex if GG is different from a path, a claw K1,3K1,3 and a cycle. As an application we prove that W(Li(T))≠W(T)W(Li(T))≠W(T) for every i≥3i≥3 if TT is a tree in which no leaf is adjacent to a vertex of degree 2, T≠K1T≠K1 and T≠K2T≠K2. This is the first in a series of papers which resolve the question, whether there exists a tree TT for which W(Lk(T))=W(L(T))W(Lk(T))=W(L(T)) for some k≥3k≥3, which was posed in [A.A. Dobrynin, R. Entringer, I. Gutman, Wiener index of trees: Theory and applications, Acta Appl. Math. 66 (2001), 211–249].