On a conjecture about the ratio of Wiener index in iterated line graphs

Abstract

Let G be a graph. Denote by W(G) its Wiener index and denote by Lⁱ(G) its i-iterated line graph. Dobrynin and Mel’nikov proposed to estimate the extremal values for the ratio R_k(G) = W(L^k(G)) / W(G) for k ≥ 1. Motivated by this we study the ratio for higher k’s. We prove that among all trees on n vertices the path P_n has the smallest value of this ratio for k ≥ 3. We conjecture that this holds also for k = 2, and even more, for the class of all connected graphs on n vertices. Moreover, we conjecture that the maximum value of the ratio is obtained for the complete graph.