Refinement on Spectral Turán’s Theorem
Yongtao Li, Yuejian Peng
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), pp. 183–189] extended Nosal's theorem to \(K_{r+1}\)-free graphs for every integer \(r\ge 2\). This is now known as the spectral Turán theorem. Recently, Lin, Ning, and Wu [Combin. Probab. Comput., 30 (2021), pp. 258–270] proved a refinement on Nosal's theorem for nonbipartite triangle-free graphs. In this paper, we provide alternative 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-\(r\)-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.KeywordsTurán theoremspectral radiusZykov symmetrizationMSC codes05C5005C35