Trees with the second-minimal ABC energy - Xiaodi Song; Jianping Li; Jianbin Zhang; Weihua He (AIMS Mathematics, 2022)
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 $.