On Wiener index of graphs and their line graphs

Abstract : The Wiener index of a graph $G$, denoted by $W(G)$, is the sum of distances between all pairs of vertices in $G$. In this paper, we consider the relation between the Wiener index of a graph, $G$, and its line graph, $L(G)$. We show that if $G$ is of minimum degree at least two, then $W(G) ≤ W(L(G))$. We prove that for every non-negative integer g0, there exists $g > g_0$, such that there are infinitely many graphs $G$ of girth $g$, satisfying $W(G) = W(L(G))$. This partially answers a question raised by Dobrynin and Mel'nikov [8] and encourages us to conjecture that the answer to a stronger form of their question is affirmative.
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Nathann Cohen, Darko Dimitrov, Roi Krakovski, Riste Skrekovski, Vida Vukašinović. On Wiener index of graphs and their line graphs. MATCH Communications in Mathematical and in Computer Chemistry, 2010, 64 (3), pp.683-698. ⟨hal-00531288⟩

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