Abstract: The celebrated Erdős-Ko-Rado theorem states that given n ≥ 2k, every intersecting k-uniform hypergraph G on n vertices has at most $(_{k - 1}^{n - 1})$ 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 ≥ 2k. Then, the sharp upper bounds for the spectral radius of $\mathcal{A}$_α(G) and $\mathcal{Q}$^*(G) are presented, where $\mathcal{A}$_α(G) = α$\mathcal{D}$(G) + (1 - α)$\mathcal{A}$(G) is a convex linear combination of the degree diagonal tensor $\mathcal{D}$(G) and the adjacency tensor $\mathcal{A}$(G) for 0 ≤ α < 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.

Key words: Erd?s-Ko-Rado theorem, Intersecting hypergraph, Tensor, Spectral radius