The Spectral Radii of Intersecting Uniform Hypergraphs

Abstract

The celebrated Erdős–Ko–Rado theorem states that given $n\geqslant 2k,$ every intersecting k-uniform hypergraph G on n vertices has at most $\left( {\begin{array}{c}n-1\\ k-1\end{array}}\right)$ edges. This paper states spectral versions of the Erdős–Ko–Rado theorem: let G be an intersecting k-uniform hypergraph on n vertices with $n\geqslant2k.$ Then, the sharp upper bounds for the spectral radius of $\mathcal {A}_{\alpha }(G)$ and $\mathcal {Q}^{*}(G)$ are presented, where $\mathcal {A}_{\alpha }(G)=\alpha \mathcal {D}(G)+(1-\alpha ) \mathcal {A}(G)$ is a convex linear combination of the degree diagonal tensor $\mathcal {D}(G)$ and the adjacency tensor $\mathcal {A}(G)$ for $0\leqslant \alpha < 1,$ and $\mathcal {Q}^{*}(G)$ is the incidence $\mathcal {Q}$-tensor, respectively. Furthermore, when $n>2k,$ the extremal hypergraphs which attain the sharp upper bounds are characterized. The proof mainly relies on the Perron–Frobenius theorem for nonnegative tensor and the property of the maximizing connected hypergraphs.