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Graph Theory 102 (2023) 502-520] proved that for $n\\geq\\frac{13}{2}r$, if $G$ is a $B_{r+1}$-free graph of order $n$, then $\\rho(G)\\leq\\rho(T_{n,2})$, with equality if and only if $G\\cong T_{n,2}$. Note that the extremal graph $T_{n,2}$ is bipartite. Motivated by the above elegant result, we investigate the spectral Tur\\'{a}n problem of non-bipartite $B_{r+1}$-free graphs of order $n$. 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