The Laplacian of a uniform hypergraph

Shenglong Hu () and Liqun Qi ()
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Shenglong Hu: The Hong Kong Polytechnic University
Liqun Qi: The Hong Kong Polytechnic University

Journal of Combinatorial Optimization, 2015, vol. 29, issue 2, No 1, 366 pages

Abstract: Abstract 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.

Keywords: Tensor; Eigenvalue; Hypergraph; Laplacian (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (9)

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DOI: 10.1007/s10878-013-9596-x

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