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On comparing the variable Zagreb indices
 MATCH Commun. Math. Comput. Chem
, 2010
"... Abstract Let G be a simple graph with n vertices and m edges. The variable first and second Zagreb indices are defined to be where λ is any real number. In this paper, it is shown that λ M 1 (G)/n ≥ λ M 2 (G)/m for all graphs G and λ ∈ (−∞, 0), which implies the results in [6, 9, 13]. We also show ..."
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Abstract Let G be a simple graph with n vertices and m edges. The variable first and second Zagreb indices are defined to be where λ is any real number. In this paper, it is shown that λ M 1 (G)/n ≥ λ M 2 (G)/m for all graphs G and λ ∈ (−∞, 0), which implies the results in [6, 9, 13]. We also show that the relationship of numerical value between λ M 1 (G)/n and λ M 2 (G)/m is indefinite in the distinct trees (resp. chemical graphs and bicyclic graphs) for λ ∈ (1, +∞). With the conclusions in [9, 10], we finish discussing the direct comparison between λ M 1 (G)/n and λ M 2 (G)/m in trees (resp. chemical graphs) for λ ∈ R.
Some classes of graphs (dis)satisfying the Zagreb indices inequality
, 2009
"... Recently Hansen and Vukičević [10] proved that the inequality M1/n ≤ M2/m, where M1 and M2 are the first and second Zagreb indices, holds for chemical graphs, and Vukičević and Graovac [17] proved that this also holds for trees. In both works is given a distinct counterexample for which this inequal ..."
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Recently Hansen and Vukičević [10] proved that the inequality M1/n ≤ M2/m, where M1 and M2 are the first and second Zagreb indices, holds for chemical graphs, and Vukičević and Graovac [17] 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 M1/n ≤ M2/m. Namely every graph G whose degrees of vertices are in the interval [c, c + ⌈ √ c ⌉] for some integer c, satisfies this inequality. In addition, we prove that for any ∆ ≥ 5, there is an infinite family of graphs of maximum degree ∆ such that the inequality is false. Moreover, an alternative and slightly shorter proof for trees is presented, as well as for unicyclic graphs.