# On Wiener index of graphs and their line graphs

1 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , Laboratoire I3S - 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.
Document type :
Journal articles

Cited literature [18 references]

https://hal.archives-ouvertes.fr/hal-00531288
Contributor : Nathann Cohen <>
Submitted on : Tuesday, November 2, 2010 - 12:20:24 PM
Last modification on : Tuesday, May 26, 2020 - 6:50:22 PM
Document(s) archivé(s) le : Friday, December 2, 2016 - 7:30:58 AM

### File

01113.pdf
Files produced by the author(s)

### Identifiers

• HAL Id : hal-00531288, version 1

### Citation

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, Mülheim a. d. Ruhr : Institut für Strahlenchemie im Max-Planck-Institut für Kohlenforschung, 2010, 64 (3), pp.683-698. ⟨hal-00531288⟩

Record views