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    Median eigenvalues of bipartite graphs

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    For a graph GG of order nn and with eigenvalues λ1λn\lambda_1\geqslant\cdots\geqslant\lambda_n, the HL-index R(G)R(G) is defined as R(G)=max{λ(n+1)/2,λ(n+1)/2}.R(G) ={\max}\left\{|\lambda_{\lfloor(n+1)/2\rfloor}|, |\lambda_{\lceil(n+1)/2\rceil}|\right\}. We show that for every connected bipartite graph GG with maximum degree Δ3\Delta\geqslant3, R(G)Δ2R(G)\leqslant\sqrt{\Delta-2} unless GG is the the incidence graph of a projective plane of order Δ1\Delta-1. We also present an approach through graph covering to construct infinite families of bipartite graphs with large HL-index
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