Complete characterization of graphs for direct comparing Zagreb indices

The classical first and second Zagreb indices of a graph G are defined as M1(G)=∑v∈VdG(v)2 and M2(G)=∑uv∈E(G)dG(u)dG(v), where dG(v) is the degree of the vertex v of graph G. Recently, Furtula et al. (2014) studied the difference between the Zagreb indices and mentioned a problem to characterize the graphs for which M1(G)>M2(G) or M1(G)