# Graph Classes (Dis)satisfying the Zagreb Indices Inequality

2 MASCOTTE - Algorithms, simulation, combinatorics and optimization for telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
Abstract : {Recently Hansen and Vukicevic proved that the inequality $M_1/n \leq M_2/m$, where $M_1$ and $M_2$ are the first and second Zagreb indices, holds for chemical graphs, and Vukicevic and Graovac proved that this also holds for trees. In both works is given a distinct counterexample for which this inequality is false in general. Here, we present some classes of graphs with prescribed degrees, that satisfy $M_1/n \leq M_2/m$: Namely every graph $G$ whose degrees of vertices are in the interval $[c; c + \sqrt c]$ for some integer $c$ satisies this inequality. In addition, we prove that for any $\Delta \geq 5$, there is an infinite family of graphs of maximum degree $\Delta$ such that the inequality is false. Moreover, an alternative and slightly shorter proof for trees is presented, as well\ as for unicyclic graphs.
Type de document :
Article dans une revue
MATCH Communications in Mathematical and in Computer Chemistry, 2011, 65 (3), pp.647-658
Domaine :

https://hal.archives-ouvertes.fr/hal-00531290
Contributeur : Nathann Cohen <>
Soumis le : mardi 2 novembre 2010 - 12:28:11
Dernière modification le : mardi 2 novembre 2010 - 13:54:40
Document(s) archivé(s) le : vendredi 2 décembre 2016 - 02:15:08

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• HAL Id : hal-00531290, version 1

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Vesna Andova, Nathann Cohen, Riste Skrekovski. Graph Classes (Dis)satisfying the Zagreb Indices Inequality. MATCH Communications in Mathematical and in Computer Chemistry, 2011, 65 (3), pp.647-658. <hal-00531290>

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