# On Wiener index of graphs and their line graphs

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
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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MATCH Communications in Mathematical and in Computer Chemistry, 2010, 64 (3), pp.683-698
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https://hal.archives-ouvertes.fr/hal-00531288
Contributeur : Nathann Cohen <>
Soumis le : mardi 2 novembre 2010 - 12:20:24
Dernière modification le : mardi 2 novembre 2010 - 13:57:45
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• HAL Id : hal-00531288, version 1

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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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