Geodesic
Median eigenvalues of bipartite graphs
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 3, pp. 899-909.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

For a graph $G$ of order $n$ and with eigenvalues $\lambda_1\geqslant\cdots\geqslant\lambda_n$, the HL-index $R(G)$ is defined as $R(G)=\max\left\{|\lambda_{\lfloor(n+1)/2\rfloor}|,|\lambda_{\lceil (n+1)/2\rceil}|\right\}$. We show that for every connected bipartite graph $G$ with maximum degree $\Delta\geqslant 3$, $R(G)\leqslant\sqrt{\Delta-2}$ unless $G$ is the the incidence graph of a projective plane of order $\Delta-1$. We also present an approach through graph covering to construct infinite families of bipartite graphs with large HL-index.
Classification : 05C50, 05B20, 05C70
Keywords: adjacency matrix, graph eigenvalues, median eigenvalues, covers 
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     title = {Median eigenvalues of bipartite graphs},
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Mohar, Bojan; Tayfeh-Rezaie, Behruz. Median eigenvalues of bipartite graphs. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 3, pp. 899-909. http://geodesic.mathdoc.fr/item/JAC_2015__41_3_a0/