Some Classes of Graphs (dis)satisfying the Zagreb Indices Inequality
نویسندگان
چکیده
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 + d√c e] 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.
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