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

The first Zagreb index image and the second Zagreb index image of a (molecular) graph image are defined as image and image, where image denotes the degree of a vertex image in image. The AutoGraphiX system [M. Aouchiche, J.M. Bonnefoy, A. Fidahoussen, G. Caporossi, P. Hansen, L. Hiesse, J. Lacheré, 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 image (where image and image) for simple connected graphs. Hansen and Vukičević [P. Hansen, D. Vukičević, 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. Vukičević and Graovac [D. Vukičević, A. Graovac, Comparing Zagreb image and image 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 image holds for graphs with image and characterize the extremal graphs, the proof of which implies the result in [P. Hansen, D. Vukičević, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165–168]. We also obtain the result that image holds for graphs with image and image.