Comparing the Zagreb indices of the NEPS of graphs

The first and the second Zagreb indices of a graph G=(V,E) are defined as M1(G)=∑u∈VdG(u)2 and M2(G)=∑uv∈EdG(u)dG(v), where dG(u) denotes the degree of a vertex u in G. It has recently been conjectured that M1(G)/|V|⩽M2(G)/|E|. Although some counterexamples have already been found, the question of characterizing graphs for which the inequality holds is left open. We show that this inequality is preserved under the NEPS of graphs, while its opposite is preserved under the direct product of graphs.