REFINEMENT ON SPECTRAL TUR\'AN'S THEOREM

A well-known result in extremal spectral graph theory, known as Nosal's theorem,states that if Gis a triangle-free graph onnvertices, then\lambda (G)\leq \lambda (K\lfloor n2\rfloor ,\lceil n2\rceil ), equality holds if andonly ifG=K\lfloor n2\rfloor ,\lceil n2\rceil . Nikiforov [Linear Algebra Appl.,427 (2007), pp. 183--189] extended Nosal'stheorem toKr+1-free graphs for every integerr\geq 2. This is now known as the spectral Tur\'antheorem. Recently, Lin, Ning, and Wu [Combin. Probab. Comput., 30 (2021), pp. 258--270] proveda refinement on Nosal's theorem for nonbipartite triangle-free graphs. In this paper, we providealternative proofs for both the result of Nikiforov and the result of Lin, Ning, and Wu. Moreover,our new proof can allow us to extend the later result to non-r-partiteKr+1-free graphs. Our resultrefines the theorem of Nikiforov and it also can be viewed as a spectral version of a theorem ofBrouwer