Abstract: A well-known result in extremal spectral graph theory,

known as Nosal's theorem, states that if G is a triangle-free graph on n vertices, then \lambda (G) \le \lambda (K_{\lfloor \frac{n}{2}\rfloor, \lceil \frac{n}{2} \rceil }), equality holds if and only if G=K_{\lfloor \frac{n}{2}\rfloor, \lceil \frac{n}{2} \rceil }. Nikiforov [Linear Algebra Appl. 427 (2007)] extended Nosal's theorem to K_{r+1}-free graphs for every integer r\ge 2. This is known as the spectral Tur\'{a}n theorem. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)]

proved a refinement on Nosal's theorem for non-bipartite triangle-free graphs. In this talk, we provide alternative proofs for the result of Nikiforov and the result of Lin, Ning and Wu. Our proof can allow us to extend the later result to non-r-partite K_{r+1}-free graphs. Our result refines the theorem of Nikiforov and it also can be viewed as a spectral version of a theorem of Brouwer. This is a joint work with Yongtao Li.