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Nikiforov [Linear Algebra Appl., 2007] proved the spectral Tur\\'{a}n theorem, which implies that $G(r,s)$ is the $r$-partite Tur\\'{a}n graph $T_{n,r}$ for $s\\leq r$. Lin, Ning, and Wu [Combin. Probab. Comput., 2021] characterized the unique spectral extremal graph $G(2,3)$. This result was later extended by Li and Peng [SIAM J. Discrete Math., 2023] to all $s=r+1\\geq 3$. 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