The Laplacian of a uniform hypergraph

In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a \(k\)-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval \([0,2]\), and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H\(^+\)-eigenvalues of the Laplacian and all the smallest H\(^+\)-eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H\(^+\)-eigenvalues of the Laplacian that are less than one are completely characterized by the spectral components of the hypergraph and vice verse. The smallest H-eigenvalue, which is also an H\(^+\)-eigenvalue, of the Laplacian is zero. When \(k\) is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If \(k\) is odd, then its largest H-eigenvalue is always strictly less than two. The largest H\(^+\)-eigenvalue of the Laplacian for a hypergraph having at least one edge is one; and its nonnegative eigenvectors are in one to one correspondence with the flower hearts of the hypergraph. The second smallest H\(^+\)-eigenvalue of the Laplacian is positive if and only if the hypergraph is connected. The number of connected components of a hypergraph is determined by the H\(^+\)-geometric multiplicity of the zero H\(^+\)-eigenvalue of the Lapalacian.