For a connected graph $G$ on at least three vertices, the augmented Zagreb index (AZI) of $G$ is defined as $$AZI(G)=\sum_{ uv\in E(G)}\left(\frac{ d(u)d(v)}{ d(u)+d(v)-2}\right)^{ 3},$$ being a topological index well-correlated with the formation heat of heptanes and octanes. A $k$-apex tree $G$ is a connected graph admitting a $k$-subset $X\subset V(G)$ such that $G-X$ is a tree, while $G-S$ is not a tree for any $S\subset V(G)$ 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 $k\ge 4$ and $n\ge 3(k+1)$. The latter solves an open problem posed in [K. Cheng, M. Liu, F. Belardo, { \em Appl. Math. Comput.}, { \bf402} (2021), 126139].
Categories: math.CO