On the largest kth eigenvalues of trees with n≡0 (mod k) Original Research Article

We consider the only remaining unsolved case n≡0 (mod k) for the largest kth eigenvalue of trees with n vertices. In 1995, Jia-yu Shao gave complete solutions for the cases k=2,3,4,5,6 and provided some necessary conditions for extremal trees in general cases (cf. [Linear Algebra Appl. 221 (1995) 131]). We further improve Shaoʹs necessary condition in this paper, which can be seen as the continuation of [Linear Algebra Appl. 221 (1995) 131].