Spectral determination of trees with large diameter and small spectral radius

Yuan, Shao and Liu proved that the H-shape tree H0n = P1,n-6 1,2;n-3 minimizes the spectral radius among all graphs with order n 9 and diameter n-4. In this paper, we achieve the spectral characterization of all graphs in the set H0 = {H0n}n,8. More precisely we show that H0n is determined by its spectrum if and only if n =6 8, 9,12, and detect all cospectral mates of H80, H90 and H012. Divisibility between characteristic polynomials of graphs turns out to be an important tool to reach our goals.