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

机译：比较最大和最小度之间差异小的图的Zagreb索引

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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) = Sigma(u is an element of V(G))(d(u))(2) and M-2(G) = Sigma(u nu is an element of V(G))d(u)d(nu), 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: I 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 <= M-2/m (where n = vertical bar V(G)vertical bar and m = vertical bar E(G)vertical bar) 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 ID. 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 <= M-2/m holds for graphs with Delta(G) - delta(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 <= M-2/m holds for graphs with Delta(G) - delta(G) <= 3 and B(G) not equal 2.
机译：（分子）图G的第一Zagreb指数M-1（G）和第二Zagreb指数M-2（G）定义为M-1（G）= Sigma（u是V（G）的元素） （d（u））（2）和M-2（G）= Sigma（u nu是V（G））d（u）d（nu）的元素，其中d（u）表示顶点的度数AutoGraphiX系统[M. Aouchiche，J.M。Bonnefoy，A.Fidahoussen，G.Caporossi，P.Hansen，L Hiesse，J.Lachere，A.Monhait，可变邻域搜索极值图。 14. AutoGraphiX 2系统，位于：L Liberti，N。Maculan（编辑），《全局优化：从理论到实现》，施普林格，2005年; G. Caporossi，P。Hansen，极值图的可变邻域搜索：I AutoGraphiX系统，离散数学。 212（2000）29-44; G. Caporossi，P。Hansen，可变邻域搜索极值图。 5.自动寻找猜想的三种方法，离散数学。 276（2004）81-94]推测对于简单连接图，M-1 / n <= M-2 / m（其中n =垂直线V（G）垂直线，m =垂直线E（G）垂直线） 。 Hansen和Vukicevic [P. Hansen，D。Vukicevic，比较萨格勒布指数，克罗地亚。化学Acta 80（2007）165-168]证明了对化学图形是正确的，并且并非对所有图形都适用。 Vukicevic和Graovac ID。 Vukicevic，A。Graovac，比较无环分子的萨格勒布M-1和M-2指数，MATCH Commun。数学。计算化学57（2007）587-590]证明了树木也是如此。在本文中，我们证明M-1 / n <= M-2 / m对于Delta（G）-delta（G）<= 2的图成立，并刻画了极值图，其证明暗示了[ P. Hansen，D。Vukicevic，比较萨格勒布指数，克罗地亚。化学Acta 80（2007）165-168]。对于Delta（G）-delta（G）<= 3并且B（G）不等于2的图，我们还获得了M-1 / n <= M-2 / m的结果。

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