The $ϵ$-spectral radius of a connected graph is the largest eigenvalue of its eccentricity matrix. In this paper, we identify the unique $n$-vertex tree with diameter $4$ and matching number $5$ that minimizes the $ϵ$-spectral radius, and thus resolve a conjecture proposed in [W. Wei, S. Li, L. Zhang, Characterizing the extremal graphs with respect to the eccentricity spectral radius, and beyond, Discrete Math. 345 (2022) 112686].

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 $S_n$.