Improved inequality between zagreb indices of trees

Abstract

For a simple graph G with n vertices and m edges, let M1 and M2 denote the first and the second Zagreb index of G. The inequality M1/n ≤ M2/m in the case of trees has been proved first by Vukičević and Graovac [MATCH Commun. Math. Comput. Chem. 57 (2007), 587-590], and a new proof has been found recently by Andova, Cohen and Škrekovski [Ars Math. Contemp. 5 (2012), 73-76]. Here we improve this inequality by showing that, if T is not a star, then nM2 - mM 1 ≥ 2(n - 3) + (Δ - 1)(Δ - 2), where Δ is the maximum vertex degree in T.