A Note on the Maximum Value of W(L(G))/W(G)

The line graph L(G) of a graph G is defined as a graph having vertex set identical with the set of edges of G and two vertices of L(G) are adjacent if and only if the corresponding edges are incident in G. Higher iteration L i(G) is obtained by repeatedly applying the line graph operation i times. Wiener index W(G) of a graph G is defined as the sum of distances which runs over all pairs of vertices in G. The problem of establishing the extremal values and extremal graphs for the ratio W(L i(G))/W(G) was proposed by Dobrynin and Melnikov [Mathematical Chemistry Monographs, Vol. 12, 2012, pp. 85-121]. In this paper we establish the maximum value and characterize the extremal graphs for i = 1. In doing so, we derive unexpectedly an interesting relation that involves the Gutman index and the first Zagreb index.