Spectral radius and Hamiltonicity of uniform hypergraphs

Abstract

Let $n$ and $r$ be integers with $n-2\ge r\ge 3$. We prove that any $r$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with spectral radius $\hat{I}»(\mathcal{H})>\left(\genfrac{}{}{0ex}{}{n-2}{r-1}\right)$ must contain a Hamiltonian Berge cycle unless $\mathcal{H}$ is the complete graph $K_{n-1}^r$ with one additional edge. This generalizes a result proved by Fiedler and Nikiforov for graphs. As part of our proof, we show that if $|\mathcal{H}|>\left(\genfrac{}{}{0ex}{}{n-1}{r}\right)$, then $\mathcal{H}$ contains a Hamiltonian Berge cycle unless $\mathcal{H}$ is the complete graph $K_{n-1}^r$ with one additional edge, generalizing a classical theorem for graphs.