Maximal augmented Zagreb index of trees with given diameter

Abstract Let G = ( V , E ) be an n -vertex graph, where V = { v 0 , v 1 , … , v n − 1 } . The augmented Zagreb index ( A Z I ) of G is defined as A Z I ( G ) = ∑ v i v j ∈ E [ d i d j / ( d i + d j − 2 ) ] 3 , where d i is the degree of v i . Let T n d be the set of all trees on n vertices with given diameter d . In this paper, we determine the tree with maximum A Z I among T n d when n ≥ 3 2 ( d − 1 ) + 381 . Our result partially resolve a problem given in [12].