Improved Inequality between Zagreb Indices of Trees

For a simple graph G with n vertices and m edges, let M-1 and M-2 denote the first and the second Zagreb index of G. The inequality M-1/n <= M-2/m in the case of trees has been proved first by Vukicevic and Graovac [MATCH Commun. Math. Comput. Chem. 57 (2007), 587-590], and a new proof has been found recently by Andova, Cohen and Skrekovski [Ars Math. Contemp. 5 (2012), 73-76]. Here we improve this inequality by showing that, if T is not a star, then nM(2) - mM(1) >= 2(n - 3) + (Delta - 1)(Delta - 2), where Delta is the maximum vertex degree in T.