Uniform hypergraphs with the first two smallest spectral radii

TL;DR

This paper identifies the hypergraphs with the smallest and second smallest spectral radii among connected uniform hypergraphs with a given number of edges, revealing unique minimal structures.

Contribution

It establishes the uniqueness of the hypergraph structures with the smallest and second smallest spectral radii for connected uniform hypergraphs.

Findings

Loose path hypergraphs have the minimum spectral radius.

The second minimum spectral radius is uniquely achieved by specific hypergraphs.

Results apply to all connected k-uniform hypergraphs with at least one edge.

Abstract

The spectral radius of a uniform hypergraph $G$ is the the maximum modulus of the eigenvalues of the adjacency tensor of $G$. For $k\ge 2$, among connected $k$-uniform hypergraphs with $m\ge 1$ edges, we show that the $k$-uniform loose path with $m$ edges is the unique one with minimum spectral radius, and we also determine the unique ones with second minimum spectral radius when $m\ge 2$.