For r>=3, k>=1 and sufficiently large n, the unique spectral extremal graph among non-r-partite B_{r,k}-free graphs is Y_r(n), obtained from the Turán graph T_r(n) by one internal edge and some cross-edge deletions.
Erd˝ os, Some theorems on graphs, Riveon Lematematika 9 (1955) 13–17
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Spectral Tur\'an-type problem in non-$r$-partite graphs: Forbidden generalized book graph $B_{r,k}$
For r>=3, k>=1 and sufficiently large n, the unique spectral extremal graph among non-r-partite B_{r,k}-free graphs is Y_r(n), obtained from the Turán graph T_r(n) by one internal edge and some cross-edge deletions.