Maximal augmented Zagreb index of trees with given diameter

Abstract

Let G=(V,E) be an n-vertex graph, where V={v0,v1,…,vn−1}. The augmented Zagreb index (AZI) of G is defined as AZI(G)=∑vivj∈E[didj/(di+dj−2)]3, where di is the degree of vi. Let Tnd be the set of all trees on n vertices with given diameter d. In this paper, we determine the tree with maximum AZI among Tnd when n≥32(d−1)+381. Our result partially resolve a problem given in [12].