  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.
