## Connected Hypergraphs with Small Spectral Radius

- April 25, 2014
- 11 a.m.
- LeConte 312

## Abstract

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 $\rho _ r=(r-1)!\sqrt[r]{4}$. We discovered a novel method for computing the spectral radius of hypergraphs, and classified all connected r-uniform hypergraphs with spectral radius at most $\rho _ r$. In addition, we also show that the next limit point of the spectral radii of connected r-uniform hypergraphs is $\rho' _ r=(r-1)!\beta^{-\frac{1}{r}}, where \beta=-\frac{1}{6}\cdot(100+12\cdot \sqrt{69})^{\frac{1}{3}}- \frac{2}{3\cdot(100+12\cdot \sqrt{69})^{\frac{1}{3}}}+\frac{4}{3}\approx 0.2451223338$ and we classified all connected r-uniform hypergraphs with spectral radius between $\rho _ 3$ and $\rho' _ 3$. This is joint work with Professor Linyuan Lu.