Refinement on spectral Turán's theorem

Abstract: A well-known result in extremal spectral graph theory, due to Nosal and Nikiforov, 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 this result 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 this result for non-bipartite triangle-free graphs. In this paper, 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.