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.
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Notes
The matrix-tensor product is in the sense of (Qi 2005, p. 1321): \(\mathcal{L }=(l_{i_1\ldots i_k}):=P^k\cdot (\mathcal D -\mathcal{A })\) is a \(k\)-th order \(n\)-dimensional tensor with its entries being \(l_{i_1\ldots i_k}:\!=\!\sum _{j_s\in [n],\;s\in [k]}p_{i_1j_1}\cdots p_{i_kj_k}(d_{j_1\ldots j_k}-a_{j_1\ldots j_k})\).
By the discussion on (Qi 2005, p. 1315) they must appear in conjugate complex pairs. They are called N-eigenvalues in that paper.
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Acknowledgments
Liqun Qi was supported by the Hong Kong Research Grant Council (Grant No. PolyU 501909, 502510, 502111 and 501212).
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Hu, S., Qi, L. The Laplacian of a uniform hypergraph. J Comb Optim 29, 331–366 (2015). https://doi.org/10.1007/s10878-013-9596-x
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DOI: https://doi.org/10.1007/s10878-013-9596-x