The Spectral Radii of Intersecting Uniform Hypergraphs

The celebrated Erdos-Ko-Rado theorem states that given n >= 2k, every intersecting k-uniform hypergraph G on n vertices has at most ((n-1)(k-1)) edges. This paper states spectral versions of the Erdos-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 A(alpha)(G) and Q*(G) are presented, where A(alpha)(G) = alpha D(G) + (1-alpha)A(G) is a convex linear combination of the degree diagonal tensor D(G) and the adjacency tensor A(G) for 0 <= alpha < 1, and Q*(G) is the incidence 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.