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.

Funding

National Natural Science Foundation of China (CN)(11971311); National Natural Science Foundation of China(11531001)