New proofs of stability theorems on spectral graph problems
Yongtao Li, Yuejian Peng
备注：
25 pages. At the end of the paper, we proposed an open problem. Any suggestions and comments are welcome! The first author would like to thank Prof. Lihua Feng, who introduced and encouraged him to the study of fascinating spectral graph theory. Thanks also go to Prof. Vladimir Nikiforov for valuable comments and for pointing out references [51,22]
Simonovits稳定性定理和Nikiforov谱稳定性定理是极值图论中求解Turán数精确值的有力工具。最近，Füredi [J. Combin.理论系列B 115（2015）]提供了Simonovits稳定性定理的简明和当代证明。本文给出了一些极值图问题的统一处理方法，包括Nikiforov谱稳定性定理和最近由Ma和Qiu [European J. Combin. 84（2020）]。此外，还讨论了与$p$ -谱半径和无符号拉普拉斯半径有关的谱极值问题.
英文摘要：
Both the Simonovits stability theorem and the Nikiforov spectral stability theorem are powerful tools for solving exact values of Turán numbers in extremal graph theory. Recently, Füredi [J. Combin. Theory Ser. B 115 (2015)] provided a concise and contemporary proof of the Simonovits stability theorem. In this note, we present a unified treatment for some extremal graph problems, including short proofs of Nikiforov's spectral stability theorem and the clique stability theorem proved recently by Ma and Qiu [European J. Combin. 84 (2020)]. Moreover, some spectral extremal problems related to the $p$-spectral radius and signless Laplacian radius are also included.
