Connected Hypergraphs with Small Spectral Radius

In 1970 Smith classified all connected graphs with the spectral radius at most $2$. Here the spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Recently, the definition of spectral radius has been extended to $r$-uniform hypergraphs. In this paper, we generalize the Smith's theorem to $r$-uniform hypergraphs. We show that the smallest limit point of the spectral radii of connected $r$-uniform hypergraphs is $ρ_r=(r-1)!\sqrt[r]{4}$. We discovered a novel method for computing the spectral radius of hypergraphs, and classified all connected $r$-uniform hypergraphs with spectral radius at most $ρ_r$.

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