Spectral conditions for connectivity of uniform hypergraphs

Abstract A hypergraph H is connected if for every pair of vertices υ , ν ∈ V ( H ), there is a path connecting υ and ν. A connected hypergraph H is said to be k-connected (or k-verex-connected) if it has more than k vertices and remains connected whenever fewer than k vertices are deleted i.e k(H) ≥ k. H is k-edge-connected if it has at least two vertices and remains connected whenever fewer than k edges are deleted. A hypergraph H is super edge-connected if every minimum edge-cut consists of edges incident with one vertex of minimum degree. The connectivity is an important measurement for the fault tolerance of a network. One can use the above four fault-tolerant parameters of hypergraphs to analyze and describe the structure properties of hypergraphs, here, we use the hypergraph spectral theory to study its inverse problem, that is, to analyze the connectivity of hypergraphs with some spectral signatures. We characterize the k-connectivity of hypergraphs in terms of their degree sequence, and give some spectral conditions for a hypergraph to be k-connected or k-edge-connected. Furthermore, we also give some spectral conditions for a graph (hypergraph) to be connected or super-edge-connected. AMS:05C50