The geometry connectivity of hypergraphs

Abstract

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 $\beta (\mathcal{G})$.