Chebyshev's Sum Inequality and the Zagreb Indices Inequality

In a recent article, Nadeem and Siddique used Chebyshev’s sum inequality to establish the Zagreb indices inequality $M_1\mathrm{/}n\le M_2\mathrm{/}m$ for undirected graphs in the case where the degree sequence $(d_i)$ and the degree-sum sequence $(S_i)$ are similarly ordered. We show that this is actually not a completely new result and we discuss several related results that also cover similar inequalities for directed graphs, as well as sum-symmetric matrices and Eulerian directed graphs.