A Bound on the Spectral Radius of Hypergraphs with $e$ Edges

TL;DR

This paper establishes an upper bound on the spectral radius of r-uniform hypergraphs with e edges, generalizing classical graph spectral bounds and characterizing extremal structures.

Contribution

It introduces a new bound function for hypergraph spectral radius and characterizes when equality holds, extending Stanley's theorem to hypergraphs.

Findings

Spectral radius of hypergraph bounded by a specific function f_r(e).

Equality characterized by union of complete hypergraph and isolated vertices.

Generalization of Stanley's theorem from graphs to hypergraphs.

Abstract

For $r\geq 3$, let $f_r\colon [0,\infty)\to [1,\infty)$ be the unique analytic function such that $f_r({k\choose r})={k-1\choose r-1}$ for any $k\geq r-1$. We prove that the spectral radius of an $r$-uniform hypergraph $H$ with $e$ edges is at most $f_r(e)$. The equality holds if and only if $e={k\choose r}$ for some positive integer $k$ and $H$ is the union of a complete $r$-uniform hypergraph $K_k^r$ and some possible isolated vertices. This result generalizes the classical Stanley's theorem on graphs.