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

Abstract

For $r\ge 3$, let $f_r:[0,\infty )\to [1,\infty )$ be the unique analytic function such that $f_r((\genfrac{}{}{0ex}{}{k}{r}))=\left(\genfrac{}{}{0ex}{}{k-1}{r-1}\right)$ for any $k\ge r-1$. 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 $e=\left(\genfrac{}{}{0ex}{}{k}{r}\right)$ 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.

16 pages