On Wiener index of graphs and their line graphs

Nathann Cohen; Darko Dimitrov; Roi Krakovski; Riste Skrekovski; Vida Vukašinović

hal-00531288, version 1

On Wiener index of graphs and their line graphs

Nathann Cohen () ¹, Darko Dimitrov () ², Roi Krakovski () ³, Riste Skrekovski ⁴, Vida Vukašinović () ⁵

MATCH Communications in Mathematical and in Computer Chemistry 64, 3 (2010) 683-698

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.

1: MASCOTTE (INRIA Sophia Antipolis / Laboratoire I3S)
INRIA – Université Nice Sophia Antipolis [UNS] – CNRS : UMR7271
2: Institut für Informatik [Berlin]
Freie Universität Berlin
3: Department of Computer Science
Ben-Gurion University of the Negev
4: Faculty of Mathematics and Physics [Ljubljana] (FMF)
University of Ljubljana
5: Computer Systems department [Ljubljana]
Jozef Stefan Institute

hal-00531288, version 1
http://hal.archives-ouvertes.fr/hal-00531288
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From: Nathann Cohen <>
Submitted on: Tuesday, 2 November 2010 12:20:24
Updated on: Tuesday, 2 November 2010 13:57:45

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