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Title: The Laplacian of a uniform hypergraph
Authors: Hu, S
Qi, L 
Keywords: Eigenvalue
Issue Date: 2013
Source: Journal of combinatorial optimization, 2015, v. 29, no. 2, p. 331-366 How to cite?
Journal: Journal of Combinatorial Optimization 
Abstract: In this paper, we investigate the Laplacian, i.e., the normalized Laplacian tensor of a {Mathematical expression}-uniform hypergraph. We show that the real parts of all the eigenvalues of the Laplacian are in the interval {Mathematical expression}, and the real part is zero (respectively two) if and only if the eigenvalue is zero (respectively two). All the H {Mathematical expression}-eigenvalues of the Laplacian and all the smallest H {Mathematical expression}-eigenvalues of its sub-tensors are characterized through the spectral radii of some nonnegative tensors. All the H {Mathematical expression}-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 {Mathematical expression}-eigenvalue, of the Laplacian is zero. When {Mathematical expression} is even, necessary and sufficient conditions for the largest H-eigenvalue of the Laplacian being two are given. If {Mathematical expression} is odd, then its largest H-eigenvalue is always strictly less than two. The largest H {Mathematical expression}-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 {Mathematical expression}-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 {Mathematical expression}-geometric multiplicity of the zero H {Mathematical expression}-eigenvalue of the Lapalacian.
ISSN: 1382-6905
DOI: 10.1007/s10878-013-9596-x
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