Given a graph G=(V,E), the first Zagreb index M₁ is the sum of its vertices squared degrees and the second Zagreb index M₂ is the sum of its edges products of degrees. Recently, there has been much interest in comparing M₁ and M₂ (and generalizations of them). The case of trees was handled in Vukicevic and Graovac (2008). In this paper, we consider the case of cyclic graphs and provide two best possible lower bounds on M₂-M₁ in terms of order and cyclicity of G. On the basis of some experiments with the system AutoGraphiX, it was conjectured that . This was disproved in Hansen and Vukicevic (2007) both for disconnected and for connected graphs. However, it is true for chemical graphs. We show here that it still holds for unicyclic graphs but is not true in general for graphs with a larger number of independent cycles.