The proof on the conjecture of extremal graphs for the kth eigenvalues of trees

We consider the only remaining unsolved case n≡0 (mod k) for the largest kth eigenvalue λ_k.of trees with n vertices. In this paper, the conjecture for this problem in [Shao Jia-yu, On the largest kth eignevalues of trees, Linear Algebra Appl. 221 (1995) 131] is proved and (from this) the complete solution to this problem, the best upper bound and the extremal trees of λ_k, is given in general cases above.