Abstract
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.
