Graph Classes (Dis)satisfying the Zagreb Indices Inequality

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{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.

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