## Graphs and Hypergraphs with Small Spectral Radius

- Feb. 7, 2014
- 11 a.m.
- LeConte 312

## Abstract

The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix. The connected graphs with spectral radius at most $2$ are classified by Smith in 1970: the graphs with spectral radius less than 2 are exactly the simple-laced Dykin Diagram: $A _ n$, $D _ n$, $E _ 6$, $E _ 7$, and $E _ 8$ while the graphs with spectral radius $2$ are the extended simple-laced Dykin Diagram: $\tilde A _ n$, $\tilde D _ n$, $\tilde E _ 6$, $\tilde E _ 7$, $\tilde D _ 8$. Since then, several group of researchers have studied the structures of the connected graphs with small spectral radius (slightly greater than 2). In this talk, we will give an overview of recent results. We will also show a generalization of Smith's theorem for $r$-uniform hypergraphs. We classify all $r$-uniform hypergraphs with spectral radius at most $(r-1)!\sqrt[r]{4}$. The last part is a joint work with Shoudong Man.