The geometry connectivity of hypergraphs

Let $mathcal{G}$ be a $k$-uniform hypergraph, $mathcal{L}_{mathcal{G}}$ be its Laplacian tensor. And $beta( mathcal{G})$ denotes the maximum number of linearly independent nonnegative eigenvectors of $mathcal{L}_{mathcal{G}}$ corresponding to the eigenvalue $0$. In this paper, $beta( mathcal{G})$ is called the geometry connectivity of $mathcal{G}$. We show that the number of connected components of $mathcal{G}$ equals the geometry connectivity $beta( mathcal{G})$.