Graph Classes (Dis)satisfying the Zagreb Indices Inequality

Recently Hansen and Vukicevic (11) proved that the inequality M1/n ≤ M2/m, where M1 and M2 are the first and second Zagreb indices, holds for chemical graphs, and Vukicevic and Graovac (23) proved that this also holds for trees. In both works a distinct counterex- ample is given 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 Δ ≥ 5 , there is an infinite family of connected graphs of maximum degree Δ , such that the inequality is false.

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