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When $s < k/2$, $G^{k,s}$ is always odd-bipartite. We show that $G^{k,{k \\over 2}}$ is non-odd-bipartite if and only if $G$ is non-bipartite, and find that $G^{k,{k \\over 2}}$ has the same adjacency (respectively, signless Laplacian) spectral radius as $G$. 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