Median eigenvalues of bipartite graphs

Abstract

For a graph $G$ of order $n$ and with eigenvalues $λ_1\geqslant\cdots\geqslantλ_n$, the HL-index $R(G)$ is defined as $R(G) ={\max}\left\{|λ_{\lfloor(n+1)/2\rfloor}|, |λ_{\lceil(n+1)/2\rceil}|\right\}.$ We show that for every connected bipartite graph $G$ with maximum degree $Δ\geqslant3$, $R(G)\leqslant\sqrt{Δ-2}$ unless $G$ is the the incidence graph of a projective plane of order $Δ-1$. We also present an approach through graph covering to construct infinite families of bipartite graphs with large HL-index.