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 as E AZ I(G) = uv is an element of E(G) \ d(u)d(v) / 3 , d(u) + d(v) - 2 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 subset of V(G) such that G - X is a tree, while G - S is not a tree for any S subset of 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].(c) 2023 Elsevier B.V. All rights reserved.