The k-apex trees with minimum augmented Zagreb index

For a connected graph G on at least three vertices, the augmented Zagreb index (AZI) of G is defined asAZI(G)=∑uv∈E(G)(d(u)d(v)d(u)+d(v)−2)3, being a topological index well-correlated with the formation heat of alkanes. A k-apex tree G is a connected graph admitting a k-subset X⊂V(G) such that G−X is a tree, while G−S is not a tree for any S⊂V(G) of cardinality less than k. By investigating some structural properties of k-apex trees, we identify the graphs minimizing the AZI among all k-apex trees on n vertices for k≥4 and n≥3(k+1). The latter solves an open problem posed in Cheng et al. (2021) [5].