Complete Characterization of Trees with Maximal Augmented Zagreb Index

The augmented Zagreb index (AZI) of an n-vertex graph G = (V, E) is defined as AZI(G) = Sigma(vivj is an element of E)[d(i)d(j)/(d(i) + d(j) - 2)](3), where V = {v(0), v(1), ..., v(n-1)},n > 3, and d(i) denotes the degree of vertex v(i) of G. As a variant of the well-known atom-bond connectivity index, the AZI was shown to have the best predicting ability for a variety of physicochemical properties among several tested vertex-degree-based topological indices. In 2010 Furtula et al. [J. Math. Chem. 48 (2010) 370] proposed the problem of characterizing n-vertex tree(s) with maximal AZI. In the present paper we solve this problem by proving that the n-vertex balanced double star uniquely maximizes AZI if n >= 19.