A bound on the spectral radius of hypergraphs with e edges

Abstract

For r≥3, let fr:[0,∞)→[1,∞) be the unique analytic function such that fr((kr))=(k−1r−1) for any k≥r−1. We prove that the spectral radius of an r-uniform hypergraph H with e edges is at most fr(e). The equality holds if and only if e=(kr) for some positive integer k and H is the union of a complete r-uniform hypergraph Kkr and some possible isolated vertices. This result generalizes the classical Stanley's theorem on graphs.