Hypergraphs with Spectral Radius at most $\sqrt[r]{2+\sqrt{5}}$
Received:February 06, 2018  Revised:February 06, 2018
Key Word: $r$-uniform hypergraphs, spectral radius, $\alpha$-normal
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 Author Name Affiliation E-mail Man Shoudong Tianjin University of Finance and Economics manshoudong@163.com Linyuan Lu University of South Carolina, Columbia, SC 29208, USA
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In our previous paper, we classified all $r$-uniform hypergraphs with spectral radius at most $\sqrt[r]{4}$, which directly generalizes Smith's theorem for the graph case $r=2$. It is natural to investigate the structures of the hypergraphs with spectral radius slightly beyond $\sqrt[r]{4}$. For $r=2$, the graphs with spectral radius at most $\sqrt{2+\sqrt{5}}$ are classified by [{\em Brouwer-Neumaier, Linear Algebra Appl., 1989}]. Here we consider the $r$-uniform hypergraphs $H$ with spectral radius at most $\sqrt[r]{2+\sqrt{5}}$. We show that $H$ must have a quipus-structure, which is similar to the graphs with spectral radius at most $\frac{3}{2}\sqrt{2}$ [{\em Woo-Neumaier, Graphs Combin. 2007}].