Complete solution of equation W(L3(T))=W(T)W(L3(T))=W(T) for the Wiener index of iterated line graphs of trees

Let GG be a graph. Denote by Li(G)Li(G) its ii-iterated line graph and denote by W(G)W(G) its Wiener index. In Knor et al. (in press) we show that there is an infinite class TT of trees TT satisfying W(L3(T))=W(T)W(L3(T))=W(T), which disproves a conjecture of Dobrynin and Entringer. In this paper we prove that except the trees of TT, there is no non-trivial tree TT satisfying W(L3(T))=W(T)W(L3(T))=W(T). Consequently, for a tree TT and i≥3i≥3, the equation W(Li(T))=W(T)W(Li(T))=W(T) holds if and only if T∈TT∈T and i=3i=3.