Proof of a conjecture on the ϵ-spectral radius of trees

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]