Comparing the Zagreb indices for graphs with small difference between the maximum and minimum degrees

The first Zagreb index M"1(G) and the second Zagreb index M"2(G) of a (molecular) graph G are defined as M"1(G)=@?"u"@?"V"("G")(d(u))^2 and M"2(G)=@?"u"v"@?"E"("G")d(u)d(v), where d(u) denotes the degree of a vertex u in G. The AutoGraphiX system [M. Aouchiche, J.M. Bonnefoy, A. Fidahoussen, G. Caporossi, P. Hansen, L. Hiesse, J. Lachere, A. Monhait, Variable neighborhood search for extremal graphs. 14. The AutoGraphiX 2 system, in: L. Liberti, N. Maculan (Eds.), Global Optimization: From Theory to Implementation, Springer, 2005; G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs: 1 The AutoGraphiX system, Discrete Math. 212 (2000) 29-44; G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures, Discrete Math. 276 (2004) 81-94] conjectured that M"1/n@?M"2/m (where n=|V(G)| and m=|E(G)|) for simple connected graphs. Hansen and Vukicevic [P. Hansen, D. Vukicevic, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165-168] proved that it is true for chemical graphs and it does not hold for all graphs. Vukicevic and Graovac [D. Vukicevic, A. Graovac, Comparing Zagreb M"1 and M"2 indices for acyclic molecules, MATCH Commun. Math. Comput. Chem. 57 (2007) 587-590] proved that it is also true for trees. In this paper, we show that M"1/n@?M"2/m holds for graphs with @D(G)-@d(G)@?2 and characterize the extremal graphs, the proof of which implies the result in [P. Hansen, D. Vukicevic, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165-168]. We also obtain the result that M"1/n@?M"2/m holds for graphs with @D(G)-@d(G)@?3 and @d(G) 2.