Connected Hypergraphs with Small Spectral Radius

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 ${\rho }_r$.

20 pages, fixed a missing class in theorem 2 and other small typos

Cited by in corpus (6)

• On the spectral radius of a class of non-odd-bipartite even uniform hypergraphs
• The $H$-spectrum of a generalized power hypergraph
• Some results on the spectral radii of uniform hypergraphs
• Ordering uniform supertrees by their spectral radii
• On the spectral radius of nonregular uniform hypergraphs
• The first few unicyclic and bicyclic hypergraphs with larger spectral radii