New proofs of stability theorems on spectral graph problems

TL;DR

This paper offers unified, concise proofs of key stability theorems in spectral graph theory, extending to spectral extremal problems involving p-spectral radius and signless Laplacian radius.

Contribution

It provides a unified framework for stability theorems, simplifying proofs of Nikiforov's spectral stability and recent clique stability results, and explores spectral extremal problems.

Findings

Unified treatment of stability theorems

Concise proofs of spectral stability and clique stability

Extension to spectral extremal problems involving p-spectral radius

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 $p$-spectral radius and signless Laplacian radius are also included.