Lexicographical ordering by spectral moments of trees with a prescribed diameter

Let Tn,d be the class of trees with n vertices and diameter d. In this paper, the lexicographic ordering of trees in the set by spectral moments is considered, and the last trees, in an S-order, among all trees in are characterized. Moreover, all trees in Tn,d have an S-order for d⩽3 and d⩾n-2.