New proofs of stability theorems on spectral graph problems
Abstract
Both the Simonovits stability theorem and the Nikiforov spectral stability theorem are powerful tools for solving exact values of Tur\'{a}n numbers in extremal graph theory. Recently, F\"{u}redi [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 -spectral radius and signless Laplacian radius are also included.
Cite
@article{arxiv.2203.03142,
title = {New proofs of stability theorems on spectral graph problems},
author = {Yongtao Li and Yuejian Peng},
journal= {arXiv preprint arXiv:2203.03142},
year = {2022}
}
Comments
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]