Azarbaijan Shahid Madani University Communications in Combinatorics and Optimization 2538-2128 9 4 2024 12 01 Spectral determination of trees with large diameter and small spectral radius 607 623 14632 10.22049/cco.2023.28648.1651 EN Xing Gao School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China Xuanshi Jia Haide School, Ocean University of China, Qingdao 266100, China Jianfeng Wang School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China Maurizio Brunetti Department of Mathematics and Applications, University of Naples Federico II, Naples, Italy Journal Article 2023 05 12 Yuan, Shao and Liu proved  that the H-shape tree $H'_n =P_{1,2;n-3}^{1,n-6}$ minimizes the spectral radius among all graphs with order $n\geqslant 9$ and diameter $n-4$. In this paper, we achieve the spectral characterization of all graphs in the set $\mathscr{H}' = \{ H'_n\}_{n\geqslant 8}$. More precisely we show that $H'_n$ is determined by its spectrum if and only if $n \neq 8, 9,12$, and detect all cospectral mates of $H'_8$, $H'_9$ and $H'_{12}$. Divisibility between characteristic polynomials of graphs turns out to be an important tool to reach our goals. https://comb-opt.azaruniv.ac.ir/article_14632_80563b6454db4c65b6432ad4ae62b221.pdf