Trees with the second-minimal ABC energy

The atom-bond connectivity energy (ABC energy) of an undirected graph $G$, denoted by ${\mathcal{E}}_{ABC}(G)$, is defined as the sum of the absolute values of the ABC eigenvalues of $G$. Gao and Shao [The minimum ABC energy of trees, Linear Algebra Appl., 577 (2019), 186-203] proved that the star $S_n$ is the unique tree with minimum ABC energy among all trees on $n$ vertices. In this paper, we characterize the trees with the minimum ABC energy among all trees on $n$ vertices except the star $S_n$.